[PPT] - Generation of finite groups with applications to computing PowerPoint Presentation

SLIDE 1

Generation of finite groups

with applications to computing normalisers Colva Roney-Dougal University of St Andrews 11th October 2017

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 3

Minimal generation

d(G) – minimum number of generators of a (finite) group G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 4

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 5

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 6

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 7

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 8

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 9

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 10

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3. Theorem (Burness, Liebeck & Shalev 13) G – almost simple with socle S. H – maximal subgroup of S.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 11

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3. Theorem (Burness, Liebeck & Shalev 13) G – almost simple with socle S. H – maximal subgroup of S. Then d(H) ≤ 4.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 12

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3. Theorem (Burness, Liebeck & Shalev 13) G – almost simple with socle S. H – maximal subgroup of S. Then d(H) ≤ 4. M – maximal subgroup of G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 13

Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3. Theorem (Burness, Liebeck & Shalev 13) G – almost simple with socle S. H – maximal subgroup of S. Then d(H) ≤ 4. M – maximal subgroup of G. Then d(M) ≤ 6.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation

d(G) – minimum number of generators of a (finite) group G. Theorem (Consequence of CFSG) Let G be a finite simple group. Then d(G) ≤ 2. G is almost simple if there exists a nonabelian simple S s.t. S ✂ G ≤ Aut(S). S = Soc(G) is the socle of G. Theorem (Dalla Volta & Lucchini 95) Let G be a finite almost simple group with socle S. Then d(G) ≤ 3, and d(G) = 3 if and only if d(G/S) = 3. Theorem (Burness, Liebeck & Shalev 13) G – almost simple with socle S. H – maximal subgroup of S. Then d(H) ≤ 4. M – maximal subgroup of G. Then d(M) ≤ 6.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 21

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 22

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

m(G) – minimal index of a proper subgroup of G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 23

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

m(G) – minimal index of a proper subgroup of G. Theorem (Liebeck & Shalev 96) There exist constants α and β

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 24

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

m(G) – minimal index of a proper subgroup of G. Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S,

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 25

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

m(G) – minimal index of a proper subgroup of G. Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S, 1 − α m(S) < PS(2) < 1 − β m(S).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 26

Probabilistic generation of simple groups

PG(k) – probability that k random elts of G generate G. Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then PS(2) → 1 as |S| → ∞. Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then PS(2) ≥ 53/90, with equality if and

nly if S = A6.

m(G) – minimal index of a proper subgroup of G. Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S, 1 − α m(S) < PS(2) < 1 − β m(S).

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 31

Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}. Bound is best possible: If n is even then C n/2

2

≤ Sn, and d(C n/2

2

) = n/2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}. Bound is best possible: If n is even then C n/2

2

≤ Sn, and d(C n/2

2

) = n/2. d(S3) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ Spm be a transitive p-group.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}. Bound is best possible: If n is even then C n/2

2

≤ Sn, and d(C n/2

2

) = n/2. d(S3) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ Spm be a transitive p-group. Then d(P) ≤ 1 + m−2

i=0 pi.

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}. Bound is best possible: If n is even then C n/2

2

≤ Sn, and d(C n/2

2

) = n/2. d(S3) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ Spm be a transitive p-group. Then d(P) ≤ 1 + m−2

i=0 pi.

Corollary If P ≤ Sn is a p-group, then d(P) ≤ n/2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 35

Minimal generation of permutation groups

G ≤ Sn is transitive if for all α, β ∈ {1, . . . , n} there exists g ∈ G s.t. αg = β. Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ Sn. Then d(G) ≤ max{n/2, 2}. Bound is best possible: If n is even then C n/2

2

≤ Sn, and d(C n/2

2

) = n/2. d(S3) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ Spm be a transitive p-group. Then d(P) ≤ 1 + m−2

i=0 pi.

Corollary If P ≤ Sn is a p-group, then d(P) ≤ n/2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 36

Minimal generation of transitive groups

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4

Colva Roney-Dougal University of St Andrews Generation of finite groups

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Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 40

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 41

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 42

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 43

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n. Kovacs and Newman: for each prime p there exists a constant cp

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 44

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n. Kovacs and Newman: for each prime p there exists a constant cp s.t. for all b there exists a transitive p-subgroup P ≤ Spb = Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 45

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n. Kovacs and Newman: for each prime p there exists a constant cp s.t. for all b there exists a transitive p-subgroup P ≤ Spb = Sn with d(P) > cpn/√log n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 46

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n. Kovacs and Newman: for each prime p there exists a constant cp s.t. for all b there exists a transitive p-subgroup P ≤ Spb = Sn with d(P) > cpn/√log n. Theorem (Tracey 17) Can take c = 0.92, or √ 3/2 with finitely many exceptions.

(All logs to base 2, unless otherwise stated.)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 47

Minimal generation of transitive groups

Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ Sn is transitive, n > 4 and (G, n) = (D8 ◦ D8, 8) then d(G) < n/2. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ Sn is transitive, then d(G) ≤ cn/√log n. Kovacs and Newman: for each prime p there exists a constant cp s.t. for all b there exists a transitive p-subgroup P ≤ Spb = Sn with d(P) > cpn/√log n. Theorem (Tracey 17) Can take c = 0.92, or √ 3/2 with finitely many exceptions.

(All logs to base 2, unless otherwise stated.)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 48

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 49

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 50

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 51

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 52

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 53

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 54

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 55

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 56

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 57

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n. Theorem (Holt & CMRD 12) Let G ≤ Sn be a subnormal subgroup of a primitive group.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 58

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n. Theorem (Holt & CMRD 12) Let G ≤ Sn be a subnormal subgroup of a primitive group. Then d(G) ≤ max{log n, 2}.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 59

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n. Theorem (Holt & CMRD 12) Let G ≤ Sn be a subnormal subgroup of a primitive group. Then d(G) ≤ max{log n, 2}. Bound is best possible:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 60

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n. Theorem (Holt & CMRD 12) Let G ≤ Sn be a subnormal subgroup of a primitive group. Then d(G) ≤ max{log n, 2}. Bound is best possible: Consider K = (Fm

2 , +) ✂ AGLm(2) ≤ S2m.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 61

Minimal generation of primitive groups

Let ∆ ⊆ {1, . . . , n}. If for all g ∈ G, either ∆g = ∆ or ∆g ∩ ∆ = ∅, then ∆ is a block for G. G ≤ Sn is primitive if G is transitive and all blocks have size 1 or n. Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ Sn is primitive then d(G) ≤ c log n √log log n. Theorem (Holt & CMRD 12) Let G ≤ Sn be a subnormal subgroup of a primitive group. Then d(G) ≤ max{log n, 2}. Bound is best possible: Consider K = (Fm

2 , +) ✂ AGLm(2) ≤ S2m.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 62

Random generation of permutation groups

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 63

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 64

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 65

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 66

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 67

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 68

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 69

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2. If G is transitive then dǫ(G) < 0.92n

√log n + 2(log n + log log n) + t + 2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 70

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2. If G is transitive then dǫ(G) < 0.92n

√log n + 2(log n + log log n) + t + 2.

If G is a subnormal subgroup of a primitive group

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 71

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2. If G is transitive then dǫ(G) < 0.92n

√log n + 2(log n + log log n) + t + 2.

If G is a subnormal subgroup of a primitive group, then dǫ(G) < 3 log n + 2 log log n + t + 2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 72

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2. If G is transitive then dǫ(G) < 0.92n

√log n + 2(log n + log log n) + t + 2.

If G is a subnormal subgroup of a primitive group, then dǫ(G) < 3 log n + 2 log log n + t + 2. If G is primitive then dǫ(G) < log n + log log n + t + 4.59.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 73

Random generation of permutation groups

Let dǫ(G) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ. Theorem (Various people) Let ǫ ∈ (0, 1), and let t be such that ζ(t) ≤ 1 + ǫ. Let G ≤ Sn. Then dǫ(G) < n/2 + 2(log n + log log n) + t + 2. If G is transitive then dǫ(G) < 0.92n

√log n + 2(log n + log log n) + t + 2.

If G is a subnormal subgroup of a primitive group, then dǫ(G) < 3 log n + 2 log log n + t + 2. If G is primitive then dǫ(G) < log n + log log n + t + 4.59.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 74

What is a random subgroup of Sn?

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 75

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 76

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated?

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 77

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 78

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 79

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn?

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 80

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups?

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 81

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 82

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 83

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)). Lower bound:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 84

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)). Lower bound: consider C ⌊n/2⌋

2

∼ = F⌊n/2⌋

2

< Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 85

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)). Lower bound: consider C ⌊n/2⌋

2

∼ = F⌊n/2⌋

2

< Sn, and count subspaces.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 86

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)). Lower bound: consider C ⌊n/2⌋

2

∼ = F⌊n/2⌋

2

< Sn, and count subspaces. Hence: not much difference between “random amongst subgroups” and “random amongst conjugacy classes of subgroups”.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 87

What is a random subgroup of Sn?

More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is Sn (or cyclic, or An). Uniform random amongst the subgroups of Sn? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a(n) – number of subgroups of Sn. Then 2n2(1/16+o(1)) ≤ a(n) ≤ 2n2(log2(24)/6+o(1)). Lower bound: consider C ⌊n/2⌋

2

∼ = F⌊n/2⌋

2

< Sn, and count subspaces. Hence: not much difference between “random amongst subgroups” and “random amongst conjugacy classes of subgroups”.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 88

More on random subgroups of Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 89

More on random subgroups of Sn

P – property of permutation groups.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 90

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 91

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 92

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 93

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 94

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P tends to 0 as n → ∞.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 95

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P tends to 0 as n → ∞. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 96

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P tends to 0 as n → ∞. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of Sn is at most 2bn2/√log n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 97

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P tends to 0 as n → ∞. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of Sn is at most 2bn2/√log n. Hence the proportion of subgroups of Sn that are transitive tends to 0 as n → ∞.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 98

More on random subgroups of Sn

P – property of permutation groups. If have a bound fP(n) on the number of generators of a subgroup of Sn with property P, then there are at most (n!)fP(n) < 2fP(n)n log n subgroups with P. Corollary P – property such that fP(n) <

n (log n)1+ε for ε > 0. Then the

proportion of subgroups of Sn that satisfy P tends to 0 as n → ∞. Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of Sn is at most 2bn2/√log n. Hence the proportion of subgroups of Sn that are transitive tends to 0 as n → ∞.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 99

Some speculation

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 100

Some speculation

It looks likely that a random subgroup of Sn should be

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 101

Some speculation

It looks likely that a random subgroup of Sn should be “Close” to soluble.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 102

Some speculation

It looks likely that a random subgroup of Sn should be “Close” to soluble. Have all orbits “short”.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 103

Some speculation

It looks likely that a random subgroup of Sn should be “Close” to soluble. Have all orbits “short”. Have order dominated by that of its Sylow 2-subgroup.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 104

Some speculation

It looks likely that a random subgroup of Sn should be “Close” to soluble. Have all orbits “short”. Have order dominated by that of its Sylow 2-subgroup.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 105

Computational group theory

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 106

Computational group theory

General set-up when computing with a finite permutation group:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 107

Computational group theory

General set-up when computing with a finite permutation group: Input:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 108

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 109

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 110

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 111

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 112

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 113

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 114

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity. Is interesting (but harder) to look at generic case complexity.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 115

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity. Is interesting (but harder) to look at generic case complexity. Consider a subset An of the set Bn of all possible inputs, such that |An|/|Bn| → 1 as n → ∞.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 116

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity. Is interesting (but harder) to look at generic case complexity. Consider a subset An of the set Bn of all possible inputs, such that |An|/|Bn| → 1 as n → ∞. Measure the worst-case complexity on this set.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 117

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity. Is interesting (but harder) to look at generic case complexity. Consider a subset An of the set Bn of all possible inputs, such that |An|/|Bn| → 1 as n → ∞. Measure the worst-case complexity on this set. That is, we’re allowed to ignore some groups G ≤ Sn, for each n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 118

Computational group theory

General set-up when computing with a finite permutation group: Input: A set X ⊂ Sn of generators for a group G. Output: answers to questions about G. Input size is |X|n log n, so the complexity of any given algorithm is a function of |X| and n. There are effective methods to reduce X to a “useful” set of size O(n), so complexity is a normally a function of n. Traditionally, looked at worst case complexity. Is interesting (but harder) to look at generic case complexity. Consider a subset An of the set Bn of all possible inputs, such that |An|/|Bn| → 1 as n → ∞. Measure the worst-case complexity on this set. That is, we’re allowed to ignore some groups G ≤ Sn, for each n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 119

Classes P and NP

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 120

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 121

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 122

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 123

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 124

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 125

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 126

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 127

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 128

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 129

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 130

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection: Given G, H ≤ Sn, find G ∩ H.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 131

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection: Given G, H ≤ Sn, find G ∩ H. Set stabiliser: Given G ≤ Sn, ∆ ⊆ {1, . . . n}, find G{∆}. Normaliser:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 132

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection: Given G, H ≤ Sn, find G ∩ H. Set stabiliser: Given G ≤ Sn, ∆ ⊆ {1, . . . n}, find G{∆}. Normaliser: Given G, H ≤ Sn,

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 133

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection: Given G, H ≤ Sn, find G ∩ H. Set stabiliser: Given G ≤ Sn, ∆ ⊆ {1, . . . n}, find G{∆}. Normaliser: Given G, H ≤ Sn, find NG(H).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 134

Classes P and NP

The class P consists of the problems that can be solved in time a polynomial in their input size. For permutation groups: polynomial in n. The class NP consists of the problems whose solution can be verified in time a polynomial in their input size. Strictly speaking these are decision problems, but often they are equivalent to problems with other types of answers. Some problems known to be in NP but not known to be in P: Graph isomorphism: Given two graphs Γ1, Γ2, decide if Γ1 ∼ = Γ2. Subgroup intersection: Given G, H ≤ Sn, find G ∩ H. Set stabiliser: Given G ≤ Sn, ∆ ⊆ {1, . . . n}, find G{∆}. Normaliser: Given G, H ≤ Sn, find NG(H).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 135

Progress on hard permutation group problems

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 136

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 137

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 138

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 139

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 140

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser. Theorem (Babai 2017) The graph isomorphism problem for a pair of graphs on n vertices has complexity O(2(log n)c)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 141

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser. Theorem (Babai 2017) The graph isomorphism problem for a pair of graphs on n vertices has complexity O(2(log n)c), for a fixed constant c.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 142

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser. Theorem (Babai 2017) The graph isomorphism problem for a pair of graphs on n vertices has complexity O(2(log n)c), for a fixed constant c. Helfgott 2017: Can take c = 3.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 143

Progress on hard permutation group problems

A problem P is polynomial-time reducible to a problem Q if a polynomial-time soln to Q would yield a polynomial-time soln to P. Theorem (Luks 93) Graph isomorphism is polynomial-time reducible to subgroup intersection. Subgroup intersection and set stabiliser are polynomial-time equivalent. Subgroup intersection is polynomial-time reducible to normaliser. Theorem (Babai 2017) The graph isomorphism problem for a pair of graphs on n vertices has complexity O(2(log n)c), for a fixed constant c. Helfgott 2017: Can take c = 3.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 144

Polynomial-time results for special cases

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 145

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 146

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 147

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 148

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d. Given G ≤ Sn, with G ∈ Γd,

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 149

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d. Given G ≤ Sn, with G ∈ Γd, in polynomial-time one can:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 150

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d. Given G ≤ Sn, with G ∈ Γd, in polynomial-time one can: For any ∆ ⊂ {1, . . . , n}, find G{∆}.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 151

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d. Given G ≤ Sn, with G ∈ Γd, in polynomial-time one can: For any ∆ ⊂ {1, . . . , n}, find G{∆}. For any H ≤ Sn, find G ∩ H.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 152

Polynomial-time results for special cases

Let Γd = {H ≤ Sn : every nonabelian composition factor of H is isomorphic to a subgroup of Sd}. All soluble groups lie in Γd, for all d. Theorem (Luks 93) Fix d. Given G ≤ Sn, with G ∈ Γd, in polynomial-time one can: For any ∆ ⊂ {1, . . . , n}, find G{∆}. For any H ≤ Sn, find G ∩ H. Corollary If there exists a d s.t. a generic subgroup G of Sn lies in Γd, then the set stabiliser and intersection problem are generically polynomial-time.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 153

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 166

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 167

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . ,

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 168

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 169

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 170

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 171

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 172

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 173

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 174

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H), so need to find generators for N/E ≤ Sn/2.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 175

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H), so need to find generators for N/E ≤ Sn/2. Identify E with Fn/2

2

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 176

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H), so need to find generators for N/E ≤ Sn/2. Identify E with Fn/2

2

, and H with a subspace of E.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 177

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H), so need to find generators for N/E ≤ Sn/2. Identify E with Fn/2

2

, and H with a subspace of E. Describe H by a k × n/2 matrix MH, where |H| = 2k.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 178

Normalisers of elementary abelian 2-groups

(From now on, joint work with Mun See Chang & Chris Jefferson). Fix G = Sn. Consider subgroups H ≤ E = (1, 2), (3, 4), . . . , ∼ = C n/2

2

≤ Sn. Want to find N = NSn(H) ≤ C2 ≀ Sn/2. (Assume fix(H) = ∅). These are the groups which yield the lower bound in Pyber’s count of subgroups of Sn. Similar methods work for subgroups of C n/p

p

for all primes p. Methods will also be useful for groups H that have several

rbits on which they act as Cp.

E ≤ CSn(H), so need to find generators for N/E ≤ Sn/2. Identify E with Fn/2

2

, and H with a subspace of E. Describe H by a k × n/2 matrix MH, where |H| = 2k.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 179

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 180

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1  

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 181

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  .

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 182

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by:

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 183

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 184

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E).

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 185

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 186

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N. N permutes sets of linearly dependent columns of MH.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 187

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N. N permutes sets of linearly dependent columns of MH. For all αi ∈ {1, . . . , n}, point stabilisers H(α1,α2,...,αi)

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 188

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N. N permutes sets of linearly dependent columns of MH. For all αi ∈ {1, . . . , n}, point stabilisers H(α1,α2,...,αi) can be efficiently calculated from MH.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 189

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N. N permutes sets of linearly dependent columns of MH. For all αi ∈ {1, . . . , n}, point stabilisers H(α1,α2,...,αi) can be efficiently calculated from MH. Let KH be a row rank (n/2 − k) matrix whose nullspace is H.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 190

An example

H = (1, 7)(2, 8)(5, 11), (1, 7)(2, 8)(3, 9)(6, 12), (2, 8)(3, 9)(4, 10) Then MH =   1 1 1 1 1 1 1 1 1 1   ∼   1 1 1 1 1 1 1 1 1 1  . Reduce search space by: Once the image of k = 3 points in {1, . . . , n/2} are specified under g ∈ Sn, we know xg for all x ∈ H (up to E). So we can decide whether g ∈ N. N permutes sets of linearly dependent columns of MH. For all αi ∈ {1, . . . , n}, point stabilisers H(α1,α2,...,αi) can be efficiently calculated from MH. Let KH be a row rank (n/2 − k) matrix whose nullspace is H. Then N also acts naturally on columns of KH, and similar

bservations apply.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 191

Some timings

Our methods for the normaliser of subgroups of C n/p

p

are still worst case exponential.

Colva Roney-Dougal University of St Andrews Generation of finite groups

SLIDE 192

Some timings

Our methods for the normaliser of subgroups of C n/p

p

are still worst case exponential. Red = our algorithm Black = standard algorithm. Time recorded is median over 20 random instances, with a 600 second timeout.

Colva Roney-Dougal University of St Andrews Generation of finite groups