Bidirected edge-maximality of power graphs of finite cyclic groups - - PowerPoint PPT Presentation

Bidirected edge-maximality of power graphs of finite cyclic groups

Brian Curtin1 Gholam Reza Pourgholi2

1Department of Mathematics and Statistics

University of South Florida

2School of Mathematics, Statistics and Computer Science

University of Tehran

Modern Trends in Algebraic Graph Theory Villanova University June 5th, 2014

Power Graphs

G : finite group of order n

Power Graphs

G : finite group of order n − → P (G) : directed power graph

V (G) = G − → E (G) = {(g, h) | g, h ∈ G, h ∈ g − {g}}

Power Graphs

G : finite group of order n − → P (G) : directed power graph

V (G) = G − → E (G) = {(g, h) | g, h ∈ G, h ∈ g − {g}}

bidirected edges

← → E (G) = {{g, h} | (g, h) ∈ − → E (G) and (h, g) ∈ − → E (G)}

Power Graphs

G : finite group of order n − → P (G) : directed power graph

V (G) = G − → E (G) = {(g, h) | g, h ∈ G, h ∈ g − {g}}

bidirected edges

← → E (G) = {{g, h} | (g, h) ∈ − → E (G) and (h, g) ∈ − → E (G)}

Lemma

∃ bidirected edge {g, h} iff g = h generate the same subgroup.

Power Graphs

− → P (Z6)

Power Graphs

− → P (Z6)

3

Power Graphs

− → P (Z6)

3

Power Graphs

− → P (Z6)

3 2

Power Graphs

− → P (Z6)

3 2 4

Power Graphs

− → P (Z6)

3 2 4

Power Graphs

− → P (Z6)

3 2 4

Power Graphs

− → P (Z6)

3 2 4

Power Graphs

− → P (Z6)

3 2 4 1 5

Power Graphs

− → P (Z6)

3 2 4 1 5

Power Graphs

− → P (Z6)

3 2 4 1 5

Power Graphs for groups of order 8

− → P (C8)

2 6 4 1 3 5 7

Power Graphs for groups of order 8

− → P (C8)

2 6 4 1 3 5 7

− → P (C4 × C2)

00 01 21 20 11 31 10 30

Power Graphs for groups of order 8

− → P (C8)

2 6 4 1 3 5 7

− → P (C4 × C2)

00 01 21 20 11 31 10 30

− → P (C2×C2×C2)

000 001 100 010 110 101 011 111

Power Graphs for groups of order 8

− → P (C8)

2 6 4 1 3 5 7

− → P (C4 × C2)

00 01 21 20 11 31 10 30

− → P (C2×C2×C2)

000 001 100 010 110 101 011 111

− → P (Q)

1 i

• i
• 1

j

• j

k

• k

Power Graphs for groups of order 8

− → P (C8)

2 6 4 1 3 5 7

− → P (C4 × C2)

00 01 21 20 11 31 10 30

− → P (C2×C2×C2)

000 001 100 010 110 101 011 111

− → P (Q)

1 i

• i
• 1

j

• j

k

• k

− → P (D8)

e ϕρ2 ρ2 ϕ ϕρ ϕρ3 ρ ρ3

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

g: cyclic, order o(g),

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

g: cyclic, order o(g), φ(o(g)) distinct generators

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

g: cyclic, order o(g), φ(o(g)) distinct generators g is in φ(o(g)) − 1 bidirected edges

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

g: cyclic, order o(g), φ(o(g)) distinct generators g is in φ(o(g)) − 1 bidirected edges Summing over G double counts

Bidirectional edges

Notation

• (g) : order of g

φ : Euler totient

Count bidirected edges

g: cyclic, order o(g), φ(o(g)) distinct generators g is in φ(o(g)) − 1 bidirected edges Summing over G double counts

Lemma

|← → E (G)| = 1 2

• g∈G

(φ(o(g)) − 1) (1)

A group sum

Definition

φ(G) =

• g∈G

φ(o(g)). (2)

A group sum

Definition

φ(G) =

• g∈G

φ(o(g)). (2)

Corollary

|← → E (G)| = φ(G) − |G| 2 (3)

A group sum

Definition

φ(G) =

• g∈G

φ(o(g)). (2)

Corollary

|← → E (G)| = φ(G) − |G| 2 (3)

Notation

Cn : cyclic group of order n Compare φ(G), φ(Cn)

Results

Main Theorem (BC, GR Pourgholi)

Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph.

Results

Main Theorem (BC, GR Pourgholi)

Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph.

Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)

Among finite groups of given order, the cyclic group has the maximum number of edges in its directed power graph.

Results

Main Theorem (BC, GR Pourgholi)

Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph.

Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)

Among finite groups of given order, the cyclic group has the maximum number of edges in its directed power graph.

Theorem (BC, GR Pourgholi)

Among finite groups of given order, the cyclic group has the maximum number of edges in its undirected power graph.

Results restated

Main Theorem (BC, GR Pourgholi)

φ(G) ≤ φ(Cn)

Results restated

Main Theorem (BC, GR Pourgholi)

φ(G) ≤ φ(Cn) i.e.

• g∈G φ(o(g)) ≤

z∈Cn φ(o(z))

Results restated

Main Theorem (BC, GR Pourgholi)

φ(G) ≤ φ(Cn) i.e.

• g∈G φ(o(g)) ≤

z∈Cn φ(o(z))

equality iff G ∼ = Cn

Results restated

Main Theorem (BC, GR Pourgholi)

φ(G) ≤ φ(Cn) i.e.

• g∈G φ(o(g)) ≤

z∈Cn φ(o(z))

equality iff G ∼ = Cn

Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)

• g∈G o(g) ≤

z∈Cn o(z)

equality iff G ∼ = Cn

Results restated

Main Theorem (BC, GR Pourgholi)

φ(G) ≤ φ(Cn) i.e.

• g∈G φ(o(g)) ≤

z∈Cn φ(o(z))

equality iff G ∼ = Cn

Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)

• g∈G o(g) ≤

z∈Cn o(z)

equality iff G ∼ = Cn

Theorem (BC, GR Pourgholi)

• g∈G 2o(g) − φ(o(g)) ≤

z∈Cn 2o(z) − φ(o(z)),

equality iff G ∼ = Cn

φ(Cn)

Notation

n = pα1

1 pα2 2 · · · pαk k

p1 < p2 < · · · < pk primes α1, α2, . . . , αk ∈ Z+

φ(Cn)

Notation

n = pα1

1 pα2 2 · · · pαk k

p1 < p2 < · · · < pk primes α1, α2, . . . , αk ∈ Z+

Lemma

φ(Cn) =

d|n φ(d)2 = k h=1 p

2αh h

(ph−1)+2 ph+1

φ(Cn)

Notation

n = pα1

1 pα2 2 · · · pαk k

p1 < p2 < · · · < pk primes α1, α2, . . . , αk ∈ Z+

Lemma

φ(Cn) =

d|n φ(d)2 = k h=1 p

2αh h

(ph−1)+2 ph+1

Definition

Q = k

h=1 ph+1 ph−1

φ(Cn)

Notation

n = pα1

1 pα2 2 · · · pαk k

p1 < p2 < · · · < pk primes α1, α2, . . . , αk ∈ Z+

Lemma

φ(Cn) =

d|n φ(d)2 = k h=1 p

2αh h

(ph−1)+2 ph+1

Definition

Q = k

h=1 ph+1 ph−1

Lemma

φ(Cn) > n2

Q

An inequality

If G is a counter example to main theorem:

average of φ(o(g)) over G:

An inequality

If G is a counter example to main theorem:

average of φ(o(g)) over G:

φ(G) n

≥ φ(Cn)

n

> n

Q

An inequality

If G is a counter example to main theorem:

average of φ(o(g)) over G:

φ(G) n

≥ φ(Cn)

n

> n

Q

∃ g ∈ G with n < Qφ(o(g))

An inequality

If G is a counter example to main theorem:

average of φ(o(g)) over G:

φ(G) n

≥ φ(Cn)

n

> n

Q

∃ g ∈ G with n < Qφ(o(g))

Key Theorem (technical proof)

p : largest prime divisor of n If ∃ g ∈ G\{id} st n < Qφ(o(g)), (as occurs if counterexample) Then ∃ normal (unique) Sylow p-subgroup P of G. P ⊆ g, so P cyclic.

Structure

Theorem (Schur-Zassenhaus)

If K ⊳ G with (|K|, |G : K|) = 1, then

Structure

Theorem (Schur-Zassenhaus)

If K ⊳ G with (|K|, |G : K|) = 1, then G = K ⋊ϕ H (semidirect product) for some H ⊆ G and some homomorphism ϕ : H → Aut(K).

Structure

Theorem (Schur-Zassenhaus)

If K ⊳ G with (|K|, |G : K|) = 1, then G = K ⋊ϕ H (semidirect product) for some H ⊆ G and some homomorphism ϕ : H → Aut(K).

Corollary

If ∃ g ∈ G\{id} st n < Qφ(o(g)): G = P ⋊ϕ H (semidirect product) P cyclic sylow p-group H subgroup with |P|, |H| coprime.

Semidirect products

Lemma

K : finite abelian group H : finite group, (|K|, |H|) = 1

Semidirect products

Lemma

K : finite abelian group H : finite group, (|K|, |H|) = 1 ∀ k ∈ K, h ∈ H

• K⋊ϕH(kh) | oK×H(kh)

(order in semi direct product divides order in direct product)

Semidirect products

Lemma

K : finite abelian group H : finite group, (|K|, |H|) = 1 ∀ k ∈ K, h ∈ H

• K⋊ϕH(kh) | oK×H(kh)

(order in semi direct product divides order in direct product)

Corollary

φ(K ⋊ϕ H) ≤ φ(K × H).

Semidirect products

Lemma

K : finite abelian group H : finite group, (|K|, |H|) = 1 ∀ k ∈ K, h ∈ H

• K⋊ϕH(kh) | oK×H(kh)

(order in semi direct product divides order in direct product)

Corollary

φ(K ⋊ϕ H) ≤ φ(K × H).

Lemma

φ(K × H) ≤ φ(K)φ(H)

Semidirect products

Lemma

K : finite abelian group H : finite group, (|K|, |H|) = 1 ∀ k ∈ K, h ∈ H

• K⋊ϕH(kh) | oK×H(kh)

(order in semi direct product divides order in direct product)

Corollary

φ(K ⋊ϕ H) ≤ φ(K × H).

Lemma

φ(K × H) ≤ φ(K)φ(H) equality when (|K|, |H|) = 1.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2. Since J not cylic, it is dihedral group D2m.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2. Since J not cylic, it is dihedral group D2m. Cm ⋊ 0 ∼ = Cm × 0 – same cogenerators of subgroups.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2. Since J not cylic, it is dihedral group D2m. Cm ⋊ 0 ∼ = Cm × 0 – same cogenerators of subgroups. Cm ⋊ 0, Cm × 0 can’t cogenerates w/ Cm ⋊ 1, Cm × 1, resp.

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2. Since J not cylic, it is dihedral group D2m. Cm ⋊ 0 ∼ = Cm × 0 – same cogenerators of subgroups. Cm ⋊ 0, Cm × 0 can’t cogenerates w/ Cm ⋊ 1, Cm × 1, resp. Cm ⋊ 1 flips, no cogenerators, Cm × 1 has all generators of C2m

Special case

• Lemma. a, b : coprime positive integers

J := φ(Ca ⋊ϕ Cb) ≤ φ(Ca × Cb) =: H equality iff the semi-direct product is direct. Suppose equality holds. Then φ(oJ(g)) = φ(oH(g)) ∀g. Now oJ(g) = oH(g) or oJ(g) = 2oH(g) with oH(g) odd. Suppose h generates H but not J. So m = oJ(h) = n/2 is odd. Now L = h ⊳ J, |L| = m odd, |J : L| = 2. Let K be a Sylow 2-subgroup of G, so |K| = 2. Now J = LK and L ∩ K = {id}, so J = L ⋊ψ K ∼ = Cm ⋊ψ C2. Since J not cylic, it is dihedral group D2m. Cm ⋊ 0 ∼ = Cm × 0 – same cogenerators of subgroups. Cm ⋊ 0, Cm × 0 can’t cogenerates w/ Cm ⋊ 1, Cm × 1, resp. Cm ⋊ 1 flips, no cogenerators, Cm × 1 has all generators of C2m equality fails, contradiction unless s.d.p. is direct.

Outline of proof of main result

Strategy

Induct on number of prime factors.

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic.

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G|

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H)

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H) φ(H) ≤ φ(C|H|), equal iff H cyclic

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H) φ(H) ≤ φ(C|H|), equal iff H cyclic H must be cyclic

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H) φ(H) ≤ φ(C|H|), equal iff H cyclic H must be cyclic G ∼ = C|P| ⋊ϕ C|H| & Cn ∼ = C|P| × C|H|

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H) φ(H) ≤ φ(C|H|), equal iff H cyclic H must be cyclic G ∼ = C|P| ⋊ϕ C|H| & Cn ∼ = C|P| × C|H| φ(C|P| ⋊ϕ φ(C|H|) = φ(C|P| × C|H|) iff s.d.p. is direct

Outline of proof of main result

Strategy

Induct on number of prime factors. Launch with prime powers–counterexample would be cyclic. If G counterexample: G = P ⋊ϕ H P cyclic sylow p-group, (|P|, |H|) = 1, |H| fewer prime divors than |G| φ(G) ≤ φ(P × H) = φ(P)φ(H) φ(H) ≤ φ(C|H|), equal iff H cyclic H must be cyclic G ∼ = C|P| ⋊ϕ C|H| & Cn ∼ = C|P| × C|H| φ(C|P| ⋊ϕ φ(C|H|) = φ(C|P| × C|H|) iff s.d.p. is direct contradiction; no counterexamples.

A group sum inequality and its application to power graphs

To appear

Bulletin of the Australian Mathematical Society

A group sum inequality and its application to power graphs

To appear

Bulletin of the Australian Mathematical Society

ArXiv

arXiv:1311.2983

A group sum inequality and its application to power graphs

To appear

Bulletin of the Australian Mathematical Society

ArXiv

arXiv:1311.2983

Thank you