[PPT] - On probabilistic generation of PSL n ( q ) A. M. Mordcovich Joint PowerPoint Presentation

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On probabilistic generation of PSLn(q)

A. M. Mordcovich

Joint work with M. Quick, C. M. Roney-Dougal 12th of August, 2017

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Probability of generating a group

Let d(G) be the size of the smallest set that generates G.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Probability of generating a group

Let d(G) be the size of the smallest set that generates G. If we pick k elements from group G where repetitions are allowed (assuming that k ≤ d(G)), what is the probability of us generating this group?

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Probability of generating a group

Let d(G) be the size of the smallest set that generates G. If we pick k elements from group G where repetitions are allowed (assuming that k ≤ d(G)), what is the probability of us generating this group? We denote this probability by PG(k).

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Probability of generating a group

Let d(G) be the size of the smallest set that generates G. If we pick k elements from group G where repetitions are allowed (assuming that k ≤ d(G)), what is the probability of us generating this group? We denote this probability by PG(k). Example: PZ5(2)

Consider G = Z5. We aim to calculate PG(2). If we pick an element that is not the identity element, then it generates the whole group.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Probability of generating a group

Let d(G) be the size of the smallest set that generates G. If we pick k elements from group G where repetitions are allowed (assuming that k ≤ d(G)), what is the probability of us generating this group? We denote this probability by PG(k). Example: PZ5(2)

Consider G = Z5. We aim to calculate PG(2). If we pick an element that is not the identity element, then it generates the whole group. So then the only pair that does not generate the whole group is a pair of identity elements. Since the number of possible pairs is 25 we have that P2(G) = 24/25

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Definition of PG,N(k)

Let N be a normal subgroup of a group G. Let us also suppose that d(G), d(G/N) ≤ k.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Definition of PG,N(k)

Let N be a normal subgroup of a group G. Let us also suppose that d(G), d(G/N) ≤ k. If we pick k elements from G (repetitions allowed), what is the probability that they generate G given that they also generates G modulo N?

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Definition of PG,N(k)

Let N be a normal subgroup of a group G. Let us also suppose that d(G), d(G/N) ≤ k. If we pick k elements from G (repetitions allowed), what is the probability that they generate G given that they also generates G modulo N? We denote this probability by PG,N(k).

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes:

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes: Classification of Finite Simple Groups

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes: Classification of Finite Simple Groups Cyclic groups Zp of prime order

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes: Classification of Finite Simple Groups Cyclic groups Zp of prime order Alternating groups An of degree of at least 5

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes: Classification of Finite Simple Groups Cyclic groups Zp of prime order Alternating groups An of degree of at least 5 Simple groups of Lie type

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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The Classification of Finite Simple Groups

We now look at the finite simple groups and the finite almost simple groups. A group G is almost simple if it satisfies S ≤ G ≤ Aut(S) for some non-abelian simple group S. Every finite simple group lies in one of the following classes: Classification of Finite Simple Groups Cyclic groups Zp of prime order Alternating groups An of degree of at least 5 Simple groups of Lie type One of 26 sporadic simple groups

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group?

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group? Theorem For all finite simple groups G, PG(2) > 0.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group? Theorem For all finite simple groups G, PG(2) > 0. Theorem [Dixon, 1969; Kantor-Lubotzky, 1990; Liebeck-Shalev, 1995]

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group? Theorem For all finite simple groups G, PG(2) > 0. Theorem [Dixon, 1969; Kantor-Lubotzky, 1990; Liebeck-Shalev, 1995] For finite simple groups G we have PG(2) → 1 as |G| → ∞.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group? Theorem For all finite simple groups G, PG(2) > 0. Theorem [Dixon, 1969; Kantor-Lubotzky, 1990; Liebeck-Shalev, 1995] For finite simple groups G we have PG(2) → 1 as |G| → ∞. Theorem [Menezes, Quick & Roney-Dougal, 2013]

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Generation of finite simple groups.

So given a finite simple group what can we say about the probability of us picking two elements (repetition allowed) that generate the group? Theorem For all finite simple groups G, PG(2) > 0. Theorem [Dixon, 1969; Kantor-Lubotzky, 1990; Liebeck-Shalev, 1995] For finite simple groups G we have PG(2) → 1 as |G| → ∞. Theorem [Menezes, Quick & Roney-Dougal, 2013] PG(2) ≥ 53/90 = 0.588.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then .

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then PG(2) = P(x, y = G| (x, y) ∈ G × G) .

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then PG(2) = P(x, y = G| (x, y) ∈ G × G) = 1 − P(x, y = G| (x, y) ∈ G × G) .

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then PG(2) = P(x, y = G| (x, y) ∈ G × G) = 1 − P(x, y = G| (x, y) ∈ G × G) = 1 − |{(x, y) ∈ G × G| x, y = G}| |G × G| .

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then PG(2) = P(x, y = G| (x, y) ∈ G × G) = 1 − P(x, y = G| (x, y) ∈ G × G) = 1 − |{(x, y) ∈ G × G| x, y = G}| |G × G| . We notice that if x and y do not generate G if and only if they both lie in some maximal subgroup of G.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)

Let us start from the definition of PG(2) and see what we can derive from there. First let us assume that d(G) ≤ 2, then PG(2) = P(x, y = G| (x, y) ∈ G × G) = 1 − P(x, y = G| (x, y) ∈ G × G) = 1 − |{(x, y) ∈ G × G| x, y = G}| |G × G| . We notice that if x and y do not generate G if and only if they both lie in some maximal subgroup of G. {(x, y) ∈ G × G| x, y = G} =

M max G

{(x, y) ∈ M × M}.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)cont.

So to bound PG(2) we need to bound

M max G

{(x, y) ∈ M × M}

.
A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)cont.

So to bound PG(2) we need to bound

M max G

{(x, y) ∈ M × M}

.

We can use the Inclusion-Exclusion Principle to obtain both an upper bound and lower bound.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)cont.

So to bound PG(2) we need to bound

M max G

{(x, y) ∈ M × M}

.

We can use the Inclusion-Exclusion Principle to obtain both an upper bound and lower bound.

M max G

{(x, y) ∈ M × M}

≤
M max G

|M|2

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounding PG(2)cont.

So to bound PG(2) we need to bound

M max G

{(x, y) ∈ M × M}

.

We can use the Inclusion-Exclusion Principle to obtain both an upper bound and lower bound.

M max G

{(x, y) ∈ M × M}

≤
M max G

|M|2

M max G

|M|2 −

M,N max G

M=N

|M ∩ N|2 ≤

M max G

{(x, y) ∈ M × M}

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounds for PG(k)

By considering the previous and generalizing we can get the following result.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounds for PG(k)

By considering the previous and generalizing we can get the following result. Theorem Let G be a group where d(G) ≤ k, then

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Bounds for PG(k)

By considering the previous and generalizing we can get the following result. Theorem Let G be a group where d(G) ≤ k, then 1 −

M max G

|G : M|−k +

M max G

M=N

|G : M ∩ N|−k ≥ PG(k) ≥ 1 −

M max G

|G : M|−k.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Analogues for PG,N(k)

We can also derive an analogous result for PG,N(k) with a bit more effort.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Analogues for PG,N(k)

We can also derive an analogous result for PG,N(k) with a bit more effort. Theorem 1 −

M max G

NM

|G : M|−k +

M1,M2maxG

NM1,M2 M1=M2

|G : M1 ∩ M2|−k ≥ PG,N(k) ≥ 1 −

M max G

NM

|G : M|−k.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Case where G is simple

If G is simple, and M is a maximal subgroup of G then |G : M| = |G : NG(M)|.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Case where G is simple

If G is simple, and M is a maximal subgroup of G then |G : M| = |G : NG(M)|. Let M be a set of representatives for the conjugacy classes of maximal subgroups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Case where G is simple

If G is simple, and M is a maximal subgroup of G then |G : M| = |G : NG(M)|. Let M be a set of representatives for the conjugacy classes of maximal subgroups. So grouping together the conjugate maximal subgroups we can see that

M max G

|G : M|−k =

M ∈ M

|G : M|−k × |G : NG(M)| =

M ∈ M

|G : M|−(k−1).

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Case where G is simple cont.

Theorem Let G be a simple group where d(G) ≤ 2, and M be a set of representatives of the conjugacy classes of the maximal subgroups

f G then
A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Case where G is simple cont.

Theorem Let G be a simple group where d(G) ≤ 2, and M be a set of representatives of the conjugacy classes of the maximal subgroups

f G then

1 −

M ∈ M

|G : M|−1 +

M max G

M=N

|G : M ∩ N|−2 ≥ PG(2) ≥ 1 −

M ∈ M

|G : M|−1.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Maximal subgroups

From here we realise that questions asking about the probabilities PG(k) and PG,N(k) are actually questions regarding maximal subgroups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Maximal subgroups

From here we realise that questions asking about the probabilities PG(k) and PG,N(k) are actually questions regarding maximal subgroups. We have information on the maximal subgroups of simple groups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Maximal subgroups

From here we realise that questions asking about the probabilities PG(k) and PG,N(k) are actually questions regarding maximal subgroups. We have information on the maximal subgroups of simple groups. In particular, the possible maximal subgroups of Classical Simple Groups are classified into 9 Classes under Aschbacher’s Theorem.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Maximal subgroups

From here we realise that questions asking about the probabilities PG(k) and PG,N(k) are actually questions regarding maximal subgroups. We have information on the maximal subgroups of simple groups. In particular, the possible maximal subgroups of Classical Simple Groups are classified into 9 Classes under Aschbacher’s Theorem. For small dimensions we know all the the maximal subgroups for the Classical Simple Groups and their related almost-simple groups [Bray, Holt & Roney-Dougal, 2013]. Therefore we can work out bounds for the probability for these cases with relative ease.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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A theorem of Liebeck & Shalev

Theorem [Liebeck & Shalev]

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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A theorem of Liebeck & Shalev

Theorem [Liebeck & Shalev] There exist constants α, β > 0 such that 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) for all finite simple groups G. Where m(G) is the index of the largest (maximal) subgroup of G in G.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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A theorem of Liebeck & Shalev

Theorem [Liebeck & Shalev] There exist constants α, β > 0 such that 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) for all finite simple groups G. Where m(G) is the index of the largest (maximal) subgroup of G in G. Remember that 1 −

M ∈M

|G : M|−1 ≤ PG(2).

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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A theorem of Liebeck & Shalev

Theorem [Liebeck & Shalev] There exist constants α, β > 0 such that 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) for all finite simple groups G. Where m(G) is the index of the largest (maximal) subgroup of G in G. Remember that 1 −

M ∈M

|G : M|−1 ≤ PG(2). The theorem is more a statement that as |G| gets large we may get more maximal subgroups but they are dwarfed in size by the largest ones.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Consider the inequality of Liebeck and Shalev; 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) .

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Consider the inequality of Liebeck and Shalev; 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) . Our aim is to provide absolute values for α and β for specific families of groups, more specifically the Classical Simple Groups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Consider the inequality of Liebeck and Shalev; 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) . Our aim is to provide absolute values for α and β for specific families of groups, more specifically the Classical Simple Groups. The results we have obtained so far involve PSLn(q) and the related almost simple groups.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Theorem

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Theorem If G = PSL2(q) then 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) where α = 38/15 and β = 1. The left hand side becomes an equality for q = 11.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results

Theorem If G = PSL2(q) then 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) where α = 38/15 and β = 1. The left hand side becomes an equality for q = 11. If G = PSLn(q) where n > 2 then 1 − α m(G) ≤ PG(2) ≤ 1 − β m(G) where α = 57/20 and β = 16/9.The left hand side is an equality for n = 3 and q = 4. The right hand side is an equality for n = 3 and q = 3.

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results cont.

We also have lower bounds for PG,N(2) for the case of N = PSLn(q).

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results cont.

We also have lower bounds for PG,N(2) for the case of N = PSLn(q). Theorem

A. M. Mordcovich

On probabilistic generation of PSLn(q)

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Results cont.

We also have lower bounds for PG,N(2) for the case of N = PSLn(q). Theorem If G is almost simple with socle N = PSLn(q) then 1 − α m(G) ≤ PG,N(2) where α = 3983/1296 = 3.07 (2 d.p.). With equality occurring when n = 4 and q = 3, and G is the extension of PSLn(q) by the graph automorphism γ.

A. M. Mordcovich

On probabilistic generation of PSLn(q)