gap€¦ · matrix group recognition max neunhöffer introduction matrix groups constructive...

Matrix grouprecognition

Max Neunhöffer

IntroductionMatrix groups

Constructive recognition

The problemComplexity theory

Randomised algorithms

Troubles

ReductionHomomorphisms

Computing the kernel

Recursion: compositiontrees

Example: invariantsubspace

Finding reductions

Solution for leavesClassifications

Recognition of the groups

Standard generators

Verification

Status of ourimplementation

Matrix group recognition in GAP

Max Neunhöffer

University of St Andrews

15.9.2007

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Matrix groups . . .Let Fq be the field with q elements and

GLn(Fq) := {M ∈ Fn×nq | M invertible}

Given: M1, . . . , Mk ∈ GLn(Fq)

Then the Mi generate a group G ≤ GLn(Fq).

It is finite, we have |GLn(Fq)| = qn(n−1)/2 ∏ni=1(q

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

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Finding reductions

Standard generators

Verification

i − 1)

Max Neunhöffer

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Finding reductions

Standard generators

Verification

i − 1)

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |

Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?

Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?

Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?

Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups

(Maximal) subgroups?. . .

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

What do we want to determine about G?The group order |G |Membership test: Is M ∈ GLn(Fq) in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

i − 1)

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Permutation groups and matrix groups

Let n ∈ N and Sn be the symmetric group:

Sn = {π : {1, . . . , n} → {1, . . . , n} | π bijective}.

Given: π1, . . . , πk ∈ Sn

Then the πi generate a group G ≤ Sn.

It is finite, we have |Sn| = n!

Let Fq be the field with q elements and

i − 1)

Max Neunhöffer

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Permutation groups and matrix groupsLet n ∈ N and Sn be the symmetric group:

Given: π1, . . . , πk ∈ Sn

i − 1)

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Given: π1, . . . , πk ∈ Sn

i − 1)

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Given: π1, . . . , πk ∈ Sn

i − 1)

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Given: π1, . . . , πk ∈ Sn

i − 1)

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Permutation groupsLet n ∈ N and Sn be the symmetric group:

Given: π1, . . . , πk ∈ Sn

It is finite, we have |Sn| = n!.

We can determine about G algorithmically (e.g.):The group order |G |Membership test: Is M ∈ Sn in G?Homomorphisms ϕ : G → H?Kernels of homomorphisms? Is G simple?Comparison with known groups(Maximal) subgroups?. . .

Max Neunhöffer

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Matrix groups in GAP

In standard GAP:

gap> ugens;[ <an immutable 56x56 matrix over GF2>,

<an immutable 56x56 matrix over GF2> ]gap> u := Group(ugens);;gap> Size(u); time;252000341277gap> Image(NiceMonomorphism(u));<permutation group with 2 generators>

Using the upcoming genss package (with F. Noeske):

gap> Size(StabilizerChain(u)); time;2520001368

For “bigger” matrix groups both approaches do not work.

Max Neunhöffer

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Finding reductions

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In standard GAP:

Max Neunhöffer

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Finding reductions

Standard generators

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In standard GAP:

Max Neunhöffer

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Finding reductions

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Constructive recognition — first formulation

ProblemLet Fq be the field with q elements and

M1, . . . , Mk ∈ GLn(Fq).

Find for G := 〈M1, . . . , Mk 〉:The group order |G | andan algorithm that, given M ∈ GLn(Fq),

decides, whether or not M ∈ G andif so, expresses M as word in the Mi .

If this problem is solved, we call

〈M1, . . . , Mk 〉 recognised constructively.

Max Neunhöffer

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Constructive recognition — first formulation

ProblemLet Fq be the field with q elements and

M1, . . . , Mk ∈ GLn(Fq).

decides, whether or not M ∈ G andif so, expresses M as word in the Mi .

Max Neunhöffer

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Complexity of algorithms

To measure the efficiency of an algorithm, we considera class P of problems, that the algorithm can solve.

We assign to each P ∈ P its size g(P),

and prove an upper bound for the runtime L(P) of thealgorithm for P:

L(P) ≤ f (g(P))

for some function f .

The growth rate of f measures the complexity.

Example (Constructive matrix group recognition)Problem given by M1, . . . , Mk ∈ GLn(Fq).Size determined by n, k and log q.Runtime should be ≤ a polynomial in n, k and log q.

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L(P) ≤ f (g(P))

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L(P) ≤ f (g(P))

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Definition (Monte Carlo algorithms)A Monte Carlo algorithm with error probability ε is analgorithm, that is guaranteed to terminate after a finitetime, such that the probability that it returns a wrongresult is at most ε.

Definition (Las Vegas algorithm)A Las Vegas algorithm with error probability ε is analgorithm, that is guaranteed to terminate after a finitetime, such that the probability that it fails is at most ε.

Example: Comp. of |G | = 4 089 470 473 293 004 800 forpermutation group G = 〈π1, π2〉 (n = 137 632):

deterministic alg.: 112s Monte Carlo ε = 1%: 6sSaving: 95% of runtime

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

deterministic alg.: 112s

Monte Carlo ε = 1%: 6sSaving: 95% of runtime

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Constructive recognitionProblemLet Fq be the field with q elements und

M1, . . . , Mk ∈ GLn(Fq).

decides, whether or not M ∈ G, and,if so, expresses M as word in the Mi .

The runtime should be bounded from above by apolynomial in n, k and log q.A Monte Carlo Algorithmus is enough. (Verification!)

Max Neunhöffer

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M1, . . . , Mk ∈ GLn(Fq).

The runtime should be bounded from above by apolynomial in n, k and log q.

A Monte Carlo Algorithmus is enough. (Verification!)

Max Neunhöffer

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Finding reductions

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M1, . . . , Mk ∈ GLn(Fq).

The runtime should be bounded from above by apolynomial in n, k and log q.A Monte Carlo Algorithmus is enough.

(Verification!)

Max Neunhöffer

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M1, . . . , Mk ∈ GLn(Fq).

Max Neunhöffer

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Finding reductions

Standard generators

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M1, . . . , Mk ∈ GLn(Fq).

Max Neunhöffer

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Finding reductions

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Troubles

The discrete logarithm problem

If M1 = [z] ∈ F1×1q with z a primitive root of Fq. Then:

Given 0 6= [x ] ∈ F1×1q , find i ∈ N such that [x ] = [z]i .

There is no solution in polynomial time in log q known!

Integer factorisation

Some methods need a factorisation of qi − 1 for an i ≤ n.

In practice q is small ⇒ no problem.We ignore both!

Max Neunhöffer

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Troubles

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Troubles

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Troubles

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Troubles

Max Neunhöffer

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Finding reductions

Standard generators

Verification

What is a reduction?

Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

A reduction is a group homomorphism

ϕ : G → HMi 7→ Pi for all i

with the following properties:

ϕ(M) is explicitly computable for all M ∈ Gϕ is surjective: H = 〈P1, . . . , Pk 〉H is in some sense “smaller”or at least “easier to recognise constructively”e.g. H ≤ Sm or H ≤ GLn′(Fq′) with n′ log q′ < n log q

Max Neunhöffer

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Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

Max Neunhöffer

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Finding reductions

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Verification

Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

ϕ(M) is explicitly computable for all M ∈ G

ϕ is surjective: H = 〈P1, . . . , Pk 〉H is in some sense “smaller”or at least “easier to recognise constructively”e.g. H ≤ Sm or H ≤ GLn′(Fq′) with n′ log q′ < n log q

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

ϕ(M) is explicitly computable for all M ∈ Gϕ is surjective: H = 〈P1, . . . , Pk 〉

H is in some sense “smaller”or at least “easier to recognise constructively”e.g. H ≤ Sm or H ≤ GLn′(Fq′) with n′ log q′ < n log q

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

ϕ(M) is explicitly computable for all M ∈ Gϕ is surjective: H = 〈P1, . . . , Pk 〉H is in some sense “smaller”or at least “easier to recognise constructively”

e.g. H ≤ Sm or H ≤ GLn′(Fq′) with n′ log q′ < n log q

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Let G := 〈M1, . . . , Mk 〉 ≤ GLn(Fq).

Max Neunhöffer

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Finding reductions

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Computing the kernelLet ϕ : G → H be a reduction and assume that H isalready recognised constructively.

Then we can compute the kernel N of ϕ:

1 Generate a (pseudo-) random element M ∈ G,2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,3 express ϕ(M) as word in the Pi ,4 evaluate the same word in the Mi ,5 get element M ′ ∈ G with M ·M ′−1 ∈ N.6 If M is uniformly distributed in G

then M ·M ′−1 is uniformly distributed in N7 Repeat.

→ Monte Carlo algorithm to compute N

Max Neunhöffer

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Finding reductions

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Max Neunhöffer

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Finding reductions

Standard generators

Verification

1 Generate a (pseudo-) random element M ∈ G,

2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,3 express ϕ(M) as word in the Pi ,4 evaluate the same word in the Mi ,5 get element M ′ ∈ G with M ·M ′−1 ∈ N.6 If M is uniformly distributed in G

Max Neunhöffer

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Finding reductions

Standard generators

Verification

1 Generate a (pseudo-) random element M ∈ G,2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,

3 express ϕ(M) as word in the Pi ,4 evaluate the same word in the Mi ,5 get element M ′ ∈ G with M ·M ′−1 ∈ N.6 If M is uniformly distributed in G

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 Generate a (pseudo-) random element M ∈ G,2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,3 express ϕ(M) as word in the Pi ,

4 evaluate the same word in the Mi ,5 get element M ′ ∈ G with M ·M ′−1 ∈ N.6 If M is uniformly distributed in G

Max Neunhöffer

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Finding reductions

Standard generators

Verification

1 Generate a (pseudo-) random element M ∈ G,2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,3 express ϕ(M) as word in the Pi ,4 evaluate the same word in the Mi ,

5 get element M ′ ∈ G with M ·M ′−1 ∈ N.6 If M is uniformly distributed in G

Max Neunhöffer

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Finding reductions

Standard generators

Verification

1 Generate a (pseudo-) random element M ∈ G,2 map it with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,3 express ϕ(M) as word in the Pi ,4 evaluate the same word in the Mi ,5 get element M ′ ∈ G with M ·M ′−1 ∈ N.

6 If M is uniformly distributed in Gthen M ·M ′−1 is uniformly distributed in N

7 Repeat.

Max Neunhöffer

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then M ·M ′−1 is uniformly distributed in N

7 Repeat.

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Recognising image and kernel sufficesLet ϕ : G → H be a reduction and assume that both Hand the kernel N = 〈N1, . . . , Nm〉 of ϕ are alreadyrecognised constructively.

Then we have recognised G constructively:|G | = |H| · |N|. And for M ∈ GLn(Fq):

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,2 express ϕ(M) as word in the Pi ,3 evaluate the same word in the Mi ,4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,5 express M ·M ′−1 as word in the Nj ,6 get M as word in the Mi and Nj :

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

7 If M /∈ G, then at least one step does not work.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Then we have recognised G constructively:|G | = |H| · |N|.

And for M ∈ GLn(Fq):

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,

2 express ϕ(M) as word in the Pi ,3 evaluate the same word in the Mi ,4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,5 express M ·M ′−1 as word in the Nj ,6 get M as word in the Mi and Nj :

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,2 express ϕ(M) as word in the Pi ,

3 evaluate the same word in the Mi ,4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,5 express M ·M ′−1 as word in the Nj ,6 get M as word in the Mi and Nj :

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,2 express ϕ(M) as word in the Pi ,3 evaluate the same word in the Mi ,

4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,5 express M ·M ′−1 as word in the Nj ,6 get M as word in the Mi and Nj :

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,2 express ϕ(M) as word in the Pi ,3 evaluate the same word in the Mi ,4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,

5 express M ·M ′−1 as word in the Nj ,6 get M as word in the Mi and Nj :

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

1 map M with ϕ onto ϕ(M) ∈ H = 〈P1, . . . , Pk 〉,2 express ϕ(M) as word in the Pi ,3 evaluate the same word in the Mi ,4 get element M ′ ∈ G such that M ·M ′−1 ∈ N,5 express M ·M ′−1 as word in the Nj ,

6 get M as word in the Mi and Nj :M ′ =

∏in the Mi , M ·M ′−1 =

∏in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

M ′ =∏

in the Mi , M ·M ′−1 =∏

in the Nj

⇒ M =(∏

in the Nj)· (

∏in the Mi).

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Recursion: composition treesWe get a tree:

3 3N H

Up arrows: inclusionsDown arrows: homomorphisms

Old idea, substantial improvements: Seress & N. 2006

Max Neunhöffer

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Finding reductions

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Verification

N HN H

1 1 2 2

Max Neunhöffer

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Finding reductions

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Verification

N HN H

1 1 2 2

Max Neunhöffer

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Finding reductions

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Verification

Example: invariant subspaceLet V = Fn

q, then G acts on V .Let W ≤ V be an invariant subspace, i.e.:

MW = W for all M ∈ G

Choose basis (w1, . . . , wd) of W and extend to a basis

(w1, . . . , wd , wd+1, . . . , wn)

of V . After a base change the matrices in G look like this:[A B0 D

]with A ∈ Fd×d

q , B ∈ Fd×(n−d)q , D ∈ F(n−d)×(n−d)

G → GLn−d(Fq),

[A B0 D

]7→ D

is a homomorphism of groups.

Max Neunhöffer

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Finding reductions

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Verification

(w1, . . . , wd , wd+1, . . . , wn)

]with A ∈ Fd×d

q , B ∈ Fd×(n−d)q , D ∈ F(n−d)×(n−d)

G → GLn−d(Fq),

[A B0 D

]7→ D

Max Neunhöffer

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Finding reductions

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Verification

(w1, . . . , wd , wd+1, . . . , wn)

]with A ∈ Fd×d

q , B ∈ Fd×(n−d)q , D ∈ F(n−d)×(n−d)

G → GLn−d(Fq),

[A B0 D

]7→ D

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Example: invariant subspace

G → GLn−d(Fq),

[A B0 D

]7→ D

is a homomorphism of groups, its kernel is

{[A B0 D

]∈ G | D = 1

The mapping

N → GLd(Fq),

[A B0 1

]7→ A

also is a homomorphism of groups and has kernel

{[A B0 D

]∈ G | A = D = 1

This group is a p-group for q = pe:[1 B0 1

1 B′

[1 B + B′

]Together with a reduction additional information is gained!

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

G → GLn−d(Fq),

[A B0 D

]7→ D

{[A B0 D

]∈ G | D = 1

The mapping

N → GLd(Fq),

[A B0 1

]7→ A

{[A B0 D

]∈ G | A = D = 1

1 B′

[1 B + B′

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

G → GLn−d(Fq),

[A B0 D

]7→ D

{[A B0 D

]∈ G | D = 1

The mapping

N → GLd(Fq),

[A B0 1

]7→ A

{[A B0 D

]∈ G | A = D = 1

1 B′

[1 B + B′

Together with a reduction additional information is gained!

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

G → GLn−d(Fq),

[A B0 D

]7→ D

{[A B0 D

]∈ G | D = 1

The mapping

N → GLd(Fq),

[A B0 1

]7→ A

{[A B0 D

]∈ G | A = D = 1

1 B′

[1 B + B′

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

How to find reductions?

Aschbacher has defined classes C1 to C8 of subgroupsof GLn(Fq).

Theorem (Aschbacher, 1984)Let G ≤ GLn(Fq) and Z := G ∩ Z (GLn(Fq)) the subgroupof scalar matrices. Then G lies in at least one of theclasses C1 to C8 or we have:

T ⊆ G/Z ⊆ Aut(T)for a non-abelian simple group T , andG acts absolutely irreducibly on V = Fn

(This last case is called C9.)

Thus we can call in heavy artillery:the classification of finite simple groupsthe modular representation theory of simple groups

Max Neunhöffer

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Finding reductions

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How to find reductions?Aschbacher has defined classes C1 to C8 of subgroupsof GLn(Fq).

Max Neunhöffer

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Finding reductions

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Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Thus we can call in heavy artillery:the classification of finite simple groups

the modular representation theory of simple groups

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Max Neunhöffer

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Finding reductions

Standard generators

Verification

Approach for leaves of the treeIf none of the algorithms for C1 to C8 has succeeded:

1 For “small” groups compute direct isomorphism ontoa permutation group.

2 Determine, for which (simple) groupT ≤ G/Z ≤ Aut(T ) holds.

3 Find an explicit isomorphism onto a “standard copy”of an intermediate group S.

4 Finally use information about S to recognise Gconstructively.

This uses:the classification of finite simple groupsinformation about their automorphism groupsinformation about element ordersinformation about conjugacy classesclassifications of the irreducible representationsinformation about the subgroup structure

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Non-constructive recognition

Methods for non-constructive recognition:

Knowledge about representations narrows down thepossibilities

Statistics about orders of random elements

Usually this leads to Monte Carlo algorithms.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Knowledge about representations narrows down thepossibilitiesStatistics about orders of random elements

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Knowledge about representations narrows down thepossibilitiesStatistics about orders of random elements

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Standard generatorsIn G we can only multiply, invert and compute orders.

Suppose: G ∼= S with T ≤ S ≤ Aut(T ) and T simple.

Find a tuple (s1, . . . , sr ) ∈ Sr together with certain wordsp1, . . . , pm in the si , such that:

S = 〈s1, . . . , sr 〉,if (s′1, . . . , s′r ) ∈ Sr with

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

(the p′j are the same words in the s′

then si 7→ s′i for 1 ≤ i ≤ r defines an automorphismof S.

Such elements are called “standard generators” of S.

We find G ∼= S explicitly by finding a tuple (M1, . . . , Mr ) ofstandard generators in G.Often this leads to efficient Las Vegas algorithms to findexplicit isomorphisms.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Standard generatorsIn G we can only multiply, invert and compute orders.Suppose: G ∼= S with T ≤ S ≤ Aut(T ) and T simple.

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

S = 〈s1, . . . , sr 〉,

if (s′1, . . . , s′r ) ∈ Sr with|si | = |s′

i | for 1 ≤ i ≤ r ,|pj | = |p′

j | for 1 ≤ j ≤ m(the p′

j are the same words in the s′i ),

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

We find G ∼= S explicitly by finding a tuple (M1, . . . , Mr ) ofstandard generators in G.

Often this leads to efficient Las Vegas algorithms to findexplicit isomorphisms.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

|si | = |s′i | for 1 ≤ i ≤ r ,

|pj | = |p′j | for 1 ≤ j ≤ m

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Everywhere we used randomised methods:Las Vegas and Monte Carlo.

⇒ We have to check whether our result is correct!

Idea:Find (short) presentations for the leaf-groups,put these together to one for the whole group.Check the relations and thus prove the result.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Idea:Find (short) presentations for the leaf-groups,

put these together to one for the whole group.Check the relations and thus prove the result.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Idea:Find (short) presentations for the leaf-groups,put these together to one for the whole group.

Check the relations and thus prove the result.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Idea:Find (short) presentations for the leaf-groups,put these together to one for the whole group.Check the relations and thus prove the result.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

Status of our implementationWe have

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),

a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,

Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,

complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,

methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),

nearly ready non-constructive recognition,a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,

a few leaf methods,no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,

no verification.

Max Neunhöffer

Troubles

Finding reductions

Standard generators

Verification

a package recogbase providing a framework toimplement recognition algorithms and compositiontrees (Ákos Seress, N.),a package recog collecting methods to findreductions and recognise leafs constructively,Authors (currently): P. Brooksbank, M. Law,S. Linton, N., A. Niemeyer, E. O’Brien, Á. Seress,complete asymptotically best methods to handlepermutation groups,methods for most Aschbacher classes for matrixgroups and projective groups (some improvedalgorithms still needed),nearly ready non-constructive recognition,a few leaf methods,no verification.

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