Matrix group recognition: status and future? Eamonn OBrien - - PowerPoint PPT Presentation

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Matrix group recognition: status and future?

Eamonn O’Brien

University of Auckland

July 2019

Eamonn O’Brien Matrix group recognition: status and future?

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Outline of lecture

G = X ≤ GL(d, q)

Eamonn O’Brien Matrix group recognition: status and future?

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Outline of lecture

G = X ≤ GL(d, q)

◮ CompositionTree approach and status.

Eamonn O’Brien Matrix group recognition: status and future?

artlogo

Outline of lecture

G = X ≤ GL(d, q)

◮ CompositionTree approach and status. ◮ Soluble Radical model of computation: uniform approach.

Eamonn O’Brien Matrix group recognition: status and future?

artlogo

Outline of lecture

G = X ≤ GL(d, q)

◮ CompositionTree approach and status. ◮ Soluble Radical model of computation: uniform approach. ◮ Challenge problems.

Eamonn O’Brien Matrix group recognition: status and future?

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Geometry following Aschbacher

Aschbacher (1984) G maximal subgroup of GL(d, q), let V = GF(q)d be underlying vector space

Eamonn O’Brien Matrix group recognition: status and future?

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Geometry following Aschbacher

Aschbacher (1984) G maximal subgroup of GL(d, q), let V = GF(q)d be underlying vector space

◮ G preserves some natural linear structure associated with the

action of G on V , and has normal subgroup related to this structure,

Eamonn O’Brien Matrix group recognition: status and future?

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Geometry following Aschbacher

Aschbacher (1984) G maximal subgroup of GL(d, q), let V = GF(q)d be underlying vector space

◮ G preserves some natural linear structure associated with the

action of G on V , and has normal subgroup related to this structure, OR

◮ G is almost simple modulo scalars: T ≤ G/Z ≤ Aut(T)

where T is simple. e.g. G = SL(d, q).

Eamonn O’Brien Matrix group recognition: status and future?

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Geometry following Aschbacher

Aschbacher (1984) G maximal subgroup of GL(d, q), let V = GF(q)d be underlying vector space

◮ G preserves some natural linear structure associated with the

action of G on V , and has normal subgroup related to this structure, OR

◮ G is almost simple modulo scalars: T ≤ G/Z ≤ Aut(T)

where T is simple. e.g. G = SL(d, q). 7 categories giving normal subgroup

Eamonn O’Brien Matrix group recognition: status and future?

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Prototype: G acts imprimitively on V

G preserves decomposition of V as direct sum V1 ⊕ V2 ⊕ · · · ⊕ Vr

• f r > 1 subspaces of dimension s, which are permuted transitively

by G.

Eamonn O’Brien Matrix group recognition: status and future?

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Prototype: G acts imprimitively on V

G preserves decomposition of V as direct sum V1 ⊕ V2 ⊕ · · · ⊕ Vr

• f r > 1 subspaces of dimension s, which are permuted transitively

by G. Then φ : G → Sr where r|d and N = ker φ. G N = ker φ Sr = im φ Holt, Leedham-Green, O’B & Rees (1996)

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories.

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories. 2 If associated N ⊳ G exists, recognise N and G/N recursively,

ultimately obtaining composition series for the group.

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories. 2 If associated N ⊳ G exists, recognise N and G/N recursively,

ultimately obtaining composition series for the group.

3 Otherwise G is either classical group in natural representation

• r T ≤ G/Z ≤ Aut(T) for simple T.

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories. 2 If associated N ⊳ G exists, recognise N and G/N recursively,

ultimately obtaining composition series for the group.

3 Otherwise G is either classical group in natural representation

• r T ≤ G/Z ≤ Aut(T) for simple T.

◮ “Reduce" from G to (quasi)simple group L. Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories. 2 If associated N ⊳ G exists, recognise N and G/N recursively,

ultimately obtaining composition series for the group.

3 Otherwise G is either classical group in natural representation

• r T ≤ G/Z ≤ Aut(T) for simple T.

◮ “Reduce" from G to (quasi)simple group L. ◮ Name L. Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree: general strategy

G = X ≤ GL(d, q).

1 Determine (at least one of) its Aschbacher categories. 2 If associated N ⊳ G exists, recognise N and G/N recursively,

ultimately obtaining composition series for the group.

3 Otherwise G is either classical group in natural representation

• r T ≤ G/Z ≤ Aut(T) for simple T.

◮ “Reduce" from G to (quasi)simple group L. ◮ Name L. ◮ Set up “constructive isomorphisms" between L and its standard

copy.

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V .

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V . Constructively decide (at least one of) its Aschbacher categories.

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V . Constructively decide (at least one of) its Aschbacher categories. If ker φ = N ⊳ G exists, then construct both N and im φ.

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V . Constructively decide (at least one of) its Aschbacher categories. If ker φ = N ⊳ G exists, then construct both N and im φ. Glasby, Holt, Leedham-Green, Neumann, Praeger, Niemeyer, O’B, Rees, Roney-Dougal, and others: algorithms to decide deciding membership in categories.

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V . Constructively decide (at least one of) its Aschbacher categories. If ker φ = N ⊳ G exists, then construct both N and im φ. Glasby, Holt, Leedham-Green, Neumann, Praeger, Niemeyer, O’B, Rees, Roney-Dougal, and others: algorithms to decide deciding membership in categories. Membership of C1, C2, C3, C6, C8 are decidable in polynomial time; decide membership in C4, C7.

Eamonn O’Brien Matrix group recognition: status and future?

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A constructive version of Aschbacher’s theorem?

Given G = X ≤ GL(d, F) acting on V . Constructively decide (at least one of) its Aschbacher categories. If ker φ = N ⊳ G exists, then construct both N and im φ. Glasby, Holt, Leedham-Green, Neumann, Praeger, Niemeyer, O’B, Rees, Roney-Dougal, and others: algorithms to decide deciding membership in categories. Membership of C1, C2, C3, C6, C8 are decidable in polynomial time; decide membership in C4, C7. Desirable: Polynomial-time decision.

Eamonn O’Brien Matrix group recognition: status and future?

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C8 membership: Naming classical groups

Problem Given G = X ≤ GL(d, q), does G contain SX(d, q)?

Eamonn O’Brien Matrix group recognition: status and future?

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C8 membership: Naming classical groups

Problem Given G = X ≤ GL(d, q), does G contain SX(d, q)? Praeger & Neumann (1992), P & Niemeyer (1998): Monte Carlo polynomial-time algorithms to name classical group in natural repn.

Eamonn O’Brien Matrix group recognition: status and future?

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C8 membership: Naming classical groups

Problem Given G = X ≤ GL(d, q), does G contain SX(d, q)? Praeger & Neumann (1992), P & Niemeyer (1998): Monte Carlo polynomial-time algorithms to name classical group in natural repn. Search for certain kinds of ppd-elements that occur with high probability in SX(d, q) and are in only a “small" number of other subgroups of GL(d, q).

Eamonn O’Brien Matrix group recognition: status and future?

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Black box: Naming groups of Lie type

Theorem (Babai, Kantor, Palfy, Seress, 2002) Given a group G isomorphic to a simple group of Lie type of known characteristic, its standard name can be computed using a polynomial time Monte-Carlo algorithm.

Eamonn O’Brien Matrix group recognition: status and future?

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Black box: Naming groups of Lie type

Theorem (Babai, Kantor, Palfy, Seress, 2002) Given a group G isomorphic to a simple group of Lie type of known characteristic, its standard name can be computed using a polynomial time Monte-Carlo algorithm. Choose sample L of independent (nearly) uniformly distributed random elements of G.

Eamonn O’Brien Matrix group recognition: status and future?

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Black box: Naming groups of Lie type

Theorem (Babai, Kantor, Palfy, Seress, 2002) Given a group G isomorphic to a simple group of Lie type of known characteristic, its standard name can be computed using a polynomial time Monte-Carlo algorithm. Choose sample L of independent (nearly) uniformly distributed random elements of G. Find the three largest integers v1 > v2 > v3 such that a member of L has order divisible by a primitive prime divisor of one of pvi − 1.

Eamonn O’Brien Matrix group recognition: status and future?

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Black box: Naming groups of Lie type

Theorem (Babai, Kantor, Palfy, Seress, 2002) Given a group G isomorphic to a simple group of Lie type of known characteristic, its standard name can be computed using a polynomial time Monte-Carlo algorithm. Choose sample L of independent (nearly) uniformly distributed random elements of G. Find the three largest integers v1 > v2 > v3 such that a member of L has order divisible by a primitive prime divisor of one of pvi − 1. Usually {v1, v2, v3} determines |G| and name of G.

Eamonn O’Brien Matrix group recognition: status and future?

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Black box: Naming groups of Lie type

Theorem (Babai, Kantor, Palfy, Seress, 2002) Given a group G isomorphic to a simple group of Lie type of known characteristic, its standard name can be computed using a polynomial time Monte-Carlo algorithm. Choose sample L of independent (nearly) uniformly distributed random elements of G. Find the three largest integers v1 > v2 > v3 such that a member of L has order divisible by a primitive prime divisor of one of pvi − 1. Usually {v1, v2, v3} determines |G| and name of G. Altseimer & Borovik (2002): distinguish between PSp(2m, q) and Ω(2m + 1, q), q odd and m ≥ 3.

Eamonn O’Brien Matrix group recognition: status and future?

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Finding the characteristic

BKPS and other algorithms assume that input G is a simple group

• f Lie type of known characteristic.

Problem Given G ≤ GL(d, q) where G is a group of Lie type in unknown defining characteristic r. Can we determine r?

Eamonn O’Brien Matrix group recognition: status and future?

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Finding the characteristic

BKPS and other algorithms assume that input G is a simple group

• f Lie type of known characteristic.

Problem Given G ≤ GL(d, q) where G is a group of Lie type in unknown defining characteristic r. Can we determine r? Theorem (Liebeck & O’B, 2007) There is a black-box polynomial-time Monte Carlo algorithm to determine the characteristic of a quasisimple group G of Lie type, subject to the existence of an order oracle for G.

Eamonn O’Brien Matrix group recognition: status and future?

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Finding the characteristic

BKPS and other algorithms assume that input G is a simple group

• f Lie type of known characteristic.

Problem Given G ≤ GL(d, q) where G is a group of Lie type in unknown defining characteristic r. Can we determine r? Theorem (Liebeck & O’B, 2007) There is a black-box polynomial-time Monte Carlo algorithm to determine the characteristic of a quasisimple group G of Lie type, subject to the existence of an order oracle for G. Algorithm proceeds recursively through centralisers of involutions to find SL(2, Fr). Now read off r.

Eamonn O’Brien Matrix group recognition: status and future?

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Finding the characteristic

BKPS and other algorithms assume that input G is a simple group

• f Lie type of known characteristic.

Problem Given G ≤ GL(d, q) where G is a group of Lie type in unknown defining characteristic r. Can we determine r? Theorem (Liebeck & O’B, 2007) There is a black-box polynomial-time Monte Carlo algorithm to determine the characteristic of a quasisimple group G of Lie type, subject to the existence of an order oracle for G. Algorithm proceeds recursively through centralisers of involutions to find SL(2, Fr). Now read off r. Kantor & Seress (2009): version for matrix groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Constructive recognition

C = X ≤ GL(d, q) where C is (quasi)simple. C is standard copy, sometimes known as “gold copy".

Eamonn O’Brien Matrix group recognition: status and future?

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Constructive recognition

C = X ≤ GL(d, q) where C is (quasi)simple. C is standard copy, sometimes known as “gold copy". G = Y ∼ = C. Want to construct “effective" isomorphisms φ : C − → G and τ : G − → C.

Eamonn O’Brien Matrix group recognition: status and future?

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Constructive recognition

C = X ≤ GL(d, q) where C is (quasi)simple. C is standard copy, sometimes known as “gold copy". G = Y ∼ = C. Want to construct “effective" isomorphisms φ : C − → G and τ : G − → C. Key idea: use standard generators.

Eamonn O’Brien Matrix group recognition: status and future?

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Using standard generators

C = X

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Using standard generators

C = X Y = G

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Using standard generators

C = X Y = G Find S = w(X) S

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Using standard generators

C = X Y = G Find S = w(X) S Find ¯ S = w(Y ) ¯ S

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Using standard generators

C = X Y = G Find S = w(X) S Find ¯ S = w(Y ) ¯ S Define φ : C → G : S → ¯ S

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Using standard generators

C = X Y = G Find S = w(X) S Find ¯ S = w(Y ) ¯ S Define φ : C → G : S → ¯ S h = w(S) h

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Using standard generators

C = X Y = G Find S = w(X) S Find ¯ S = w(Y ) ¯ S Define φ : C → G : S → ¯ S h = w(S) h Thus ¯ h = w( ¯ S) ¯ h

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Leedham-Green and O’B, 2009; Dietrich, L-G, Lübeck, O’B, 2013; D, L-G, O’B, 2014 Theorem There is a Las Vegas algorithm that takes as input G ∼ = SX(d, q) = X and returns standard generators S for G as words in X. The algorithm has complexity O(d4 log q) measured in field operations.

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Leedham-Green and O’B, 2009; Dietrich, L-G, Lübeck, O’B, 2013; D, L-G, O’B, 2014 Theorem There is a Las Vegas algorithm that takes as input G ∼ = SX(d, q) = X and returns standard generators S for G as words in X. The algorithm has complexity O(d4 log q) measured in field operations. Theorem (Liebeck & O’B, 2016) Similar statement for exceptional groups of rank ≥ 2.

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Leedham-Green and O’B, 2009; Dietrich, L-G, Lübeck, O’B, 2013; D, L-G, O’B, 2014 Theorem There is a Las Vegas algorithm that takes as input G ∼ = SX(d, q) = X and returns standard generators S for G as words in X. The algorithm has complexity O(d4 log q) measured in field operations. Theorem (Liebeck & O’B, 2016) Similar statement for exceptional groups of rank ≥ 2. Bäärnhielm and others: Suzuki, small and large Ree groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Leedham-Green and O’B, 2009; Dietrich, L-G, Lübeck, O’B, 2013; D, L-G, O’B, 2014 Theorem There is a Las Vegas algorithm that takes as input G ∼ = SX(d, q) = X and returns standard generators S for G as words in X. The algorithm has complexity O(d4 log q) measured in field operations. Theorem (Liebeck & O’B, 2016) Similar statement for exceptional groups of rank ≥ 2. Bäärnhielm and others: Suzuki, small and large Ree groups. Key: centralisers of involutions and statistical group theory.

Eamonn O’Brien Matrix group recognition: status and future?

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Fundamental base case SL2(q)

Conder, L-G, O’B (2006): defining characteristic repns relying on discrete log oracle.

Eamonn O’Brien Matrix group recognition: status and future?

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Fundamental base case SL2(q)

Conder, L-G, O’B (2006): defining characteristic repns relying on discrete log oracle. Kantor & Kassabov (2015); Borovik & Yalçınkaya (2018): new algorithms for this task without use of such an oracle.

Eamonn O’Brien Matrix group recognition: status and future?

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Fundamental base case SL2(q)

Conder, L-G, O’B (2006): defining characteristic repns relying on discrete log oracle. Kantor & Kassabov (2015); Borovik & Yalçınkaya (2018): new algorithms for this task without use of such an oracle. Other critical base cases: SL3(q), SU3(q), Ωǫ(d, q) where d ≤ 7.

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Jambor et al. (2013): constructive recognition algorithms for An and Sn.

Eamonn O’Brien Matrix group recognition: status and future?

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Constuctive recognition algorithms

Jambor et al. (2013): constructive recognition algorithms for An and Sn. Bray, Wilson and others: standard generators and algorithms for sporadic groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Writing elements as words in standard generators

Costi, 2009; Praeger and Schneider, 2014; Cohen & Taylor, 2018. Algorithms to write elements of G as words in S.

Eamonn O’Brien Matrix group recognition: status and future?

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Verification

Theorem (Leedham-Green & O’B, 2018) Every classical group of rank r defined over GF(q) has a presentation of length O(r + log q) on its (at most 8) standard generators.

Eamonn O’Brien Matrix group recognition: status and future?

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Verification

Theorem (Leedham-Green & O’B, 2018) Every classical group of rank r defined over GF(q) has a presentation of length O(r + log q) on its (at most 8) standard generators. Liebeck & O’B (2016): use reduced Curtis-Steinberg-Tits presentations for exceptional groups. Bray et al.: Presentations on standard generators for sporadic groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Verification

Theorem (Leedham-Green & O’B, 2018) Every classical group of rank r defined over GF(q) has a presentation of length O(r + log q) on its (at most 8) standard generators. Liebeck & O’B (2016): use reduced Curtis-Steinberg-Tits presentations for exceptional groups. Bray et al.: Presentations on standard generators for sporadic groups. Explicit presentations evaluated on standard generators used to upgrade Monte Carlo algorithms to Las Vegas.

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree

Bäärnhielm, Holt, Leedham-Green, O’B (2014): algorithm which exploits geometry and constructive recognition to construct composition series (and more) for G.

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree

Bäärnhielm, Holt, Leedham-Green, O’B (2014): algorithm which exploits geometry and constructive recognition to construct composition series (and more) for G. H K I

Eamonn O’Brien Matrix group recognition: status and future?

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CompositionTree

Bäärnhielm, Holt, Leedham-Green, O’B (2014): algorithm which exploits geometry and constructive recognition to construct composition series (and more) for G. H K I

◮ Node: section H of G. ◮ Image I: image under homomorphism or isomorphism. Images

usually correspond to Aschbacher category, but also others e.g determinant map.

◮ Kernel K. ◮ Leaf is composition factor of G.

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order.

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order. If node H is not a leaf, construct recursively subtree rooted at I, then subtree rooted at K.

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order. If node H is not a leaf, construct recursively subtree rooted at I, then subtree rooted at K. H I1

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order. If node H is not a leaf, construct recursively subtree rooted at I, then subtree rooted at K. H I1 H I1 I2

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order. If node H is not a leaf, construct recursively subtree rooted at I, then subtree rooted at K. H I1 H I1 I2 H I1 K2 I2

Eamonn O’Brien Matrix group recognition: status and future?

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Tree is constructed in right depth-first order. If node H is not a leaf, construct recursively subtree rooted at I, then subtree rooted at K. H I1 H I1 I2 H I1 K2 I2 H K1 I1 K2 I2

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ.

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ. H K I

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ. H K I Sometimes easy to obtain theoretically generating sets for ker φ. e.g. Smaller Field, Semilinear, normaliser of symplectic-type group.

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ. H K I Sometimes easy to obtain theoretically generating sets for ker φ. e.g. Smaller Field, Semilinear, normaliser of symplectic-type group. Two approaches to construct kernel.

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ. H K I Sometimes easy to obtain theoretically generating sets for ker φ. e.g. Smaller Field, Semilinear, normaliser of symplectic-type group. Two approaches to construct kernel.

◮ Random generation

Eamonn O’Brien Matrix group recognition: status and future?

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Constructing kernels

Assume φ : H − → I where K = ker φ. H K I Sometimes easy to obtain theoretically generating sets for ker φ. e.g. Smaller Field, Semilinear, normaliser of symplectic-type group. Two approaches to construct kernel.

◮ Random generation ◮ Use presentation for image to construct normal generators for

kernel

Eamonn O’Brien Matrix group recognition: status and future?

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Output of CompositionTree

Given G = X ≤ GL(d, q) as input. Output:

◮ a composition series: 1 = G0 ⊳ G1 ⊳ G2 · · · ⊳ Gm = G. ◮ A representation Sk = Xk of Gk/Gk−1 ◮ Effective maps τk : Gk → Sk, φk : Sk → Gk

τk epimorphism with kernel Gk−1

◮ Map to write g ∈ G as word in X.

Eamonn O’Brien Matrix group recognition: status and future?

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Output of CompositionTree

Given G = X ≤ GL(d, q) as input. Output:

◮ a composition series: 1 = G0 ⊳ G1 ⊳ G2 · · · ⊳ Gm = G. ◮ A representation Sk = Xk of Gk/Gk−1 ◮ Effective maps τk : Gk → Sk, φk : Sk → Gk

τk epimorphism with kernel Gk−1

◮ Map to write g ∈ G as word in X.

Construct presentation for group defined by tree and verify that G satisfies the relations.

Eamonn O’Brien Matrix group recognition: status and future?

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Output of CompositionTree

Given G = X ≤ GL(d, q) as input. Output:

◮ a composition series: 1 = G0 ⊳ G1 ⊳ G2 · · · ⊳ Gm = G. ◮ A representation Sk = Xk of Gk/Gk−1 ◮ Effective maps τk : Gk → Sk, φk : Sk → Gk

τk epimorphism with kernel Gk−1

◮ Map to write g ∈ G as word in X.

Construct presentation for group defined by tree and verify that G satisfies the relations. Hence construction of tree is Las Vegas algorithm.

Eamonn O’Brien Matrix group recognition: status and future?

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Output of CompositionTree

Given G = X ≤ GL(d, q) as input. Output:

◮ a composition series: 1 = G0 ⊳ G1 ⊳ G2 · · · ⊳ Gm = G. ◮ A representation Sk = Xk of Gk/Gk−1 ◮ Effective maps τk : Gk → Sk, φk : Sk → Gk

τk epimorphism with kernel Gk−1

◮ Map to write g ∈ G as word in X.

Construct presentation for group defined by tree and verify that G satisfies the relations. Hence construction of tree is Las Vegas algorithm. CompositionTree data structure: allows membership testing etc.

Eamonn O’Brien Matrix group recognition: status and future?

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Composition factors in polynomial-time?

Eamonn O’Brien Matrix group recognition: status and future?

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Composition factors in polynomial-time?

Holt, Leedham-Green, O’B (2019) Theorem Subject to the existence of a discrete log oracle, and an integer factorisation oracle, there is a polynomial-time Monte Carlo algorithm that takes as input G := X ≤ GL(d, q) and constructs its composition factors.

Eamonn O’Brien Matrix group recognition: status and future?

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Composition factors in polynomial-time?

Holt, Leedham-Green, O’B (2019) Theorem Subject to the existence of a discrete log oracle, and an integer factorisation oracle, there is a polynomial-time Monte Carlo algorithm that takes as input G := X ≤ GL(d, q) and constructs its composition factors. Can be upgraded to Las Vegas in all cases where group has no composition factor 2G2(q).

Eamonn O’Brien Matrix group recognition: status and future?

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What can we do with outcome?

Eamonn O’Brien Matrix group recognition: status and future?

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What can we do with outcome?

Use it as infrastructure for Soluble Radical model of computation: uniform approach to computations with linear groups.

Eamonn O’Brien Matrix group recognition: status and future?

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What can we do with outcome?

Use it as infrastructure for Soluble Radical model of computation: uniform approach to computations with linear groups. Analogue to use of Schreier-Sims data structure for permutation groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G O∞(G) = largest soluble normal subgroup of G, soluble radical

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G O∞(G) = largest soluble normal subgroup of G, soluble radical S∗(G)/O∞(G) = Socle (G/O∞(G)) = T1 × . . . × Tk where Ti non-abelian simple

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G O∞(G) = largest soluble normal subgroup of G, soluble radical S∗(G)/O∞(G) = Socle (G/O∞(G)) = T1 × . . . × Tk where Ti non-abelian simple φ : G − → Sym(k) is repn of G induced by conjugation on {T1, . . . , Tk} and P(G) = ker φ

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G O∞(G) = largest soluble normal subgroup of G, soluble radical S∗(G)/O∞(G) = Socle (G/O∞(G)) = T1 × . . . × Tk where Ti non-abelian simple φ : G − → Sym(k) is repn of G induced by conjugation on {T1, . . . , Tk} and P(G) = ker φ P(G)/S∗(G) ≤ Out(T1) × . . . × Out(Tk) and so is soluble

Eamonn O’Brien Matrix group recognition: status and future?

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Characteristic structure

Finite G has characteristic series C of subgroups: 1 ≤ O∞(G) ≤ S∗(G) ≤ P(G) ≤ G O∞(G) = largest soluble normal subgroup of G, soluble radical S∗(G)/O∞(G) = Socle (G/O∞(G)) = T1 × . . . × Tk where Ti non-abelian simple φ : G − → Sym(k) is repn of G induced by conjugation on {T1, . . . , Tk} and P(G) = ker φ P(G)/S∗(G) ≤ Out(T1) × . . . × Out(Tk) and so is soluble G/P(G) ≤ Sym(k)

Eamonn O’Brien Matrix group recognition: status and future?

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Exploiting the characteristic series C

Cannon, Holt et al. (2000s): use C in practical algorithms. 1 ≤ L := O∞(G) ≤ S∗(G) ≤ P(G) ≤ G

Eamonn O’Brien Matrix group recognition: status and future?

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Exploiting the characteristic series C

Cannon, Holt et al. (2000s): use C in practical algorithms. 1 ≤ L := O∞(G) ≤ S∗(G) ≤ P(G) ≤ G Also compute series 1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ⊳ G where Ni/Ni−1 is elementary abelian.

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian.

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian. Given a problem:

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian. Given a problem: Solve problem first in G/L = G/Nr, and then, successively, solve it in G/Ni, for i = r − 1, . . . , 0.

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian. Given a problem: Solve problem first in G/L = G/Nr, and then, successively, solve it in G/Ni, for i = r − 1, . . . , 0. H := G/L has trivial Fitting subgroup. So H has a socle S which is direct product of non-abelian simple groups Ti and these are permuted under conjugation by H.

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian. Given a problem: Solve problem first in G/L = G/Nr, and then, successively, solve it in G/Ni, for i = r − 1, . . . , 0. H := G/L has trivial Fitting subgroup. So H has a socle S which is direct product of non-abelian simple groups Ti and these are permuted under conjugation by H. Problem may have nice solution for H.

Eamonn O’Brien Matrix group recognition: status and future?

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The Soluble Radical model

1 = N0 ⊳ N1 ⊳ · · · ⊳ Nr = L ≤ S∗(G) ≤ P(G) ≤ G where Ni G and Ni/Ni−1 is elementary abelian. Given a problem: Solve problem first in G/L = G/Nr, and then, successively, solve it in G/Ni, for i = r − 1, . . . , 0. H := G/L has trivial Fitting subgroup. So H has a socle S which is direct product of non-abelian simple groups Ti and these are permuted under conjugation by H. Problem may have nice solution for H. In many cases, easy to reduce the computation for TF-group H to almost simple groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Examples of algorithms using Soluble Radical model

◮ Determine conjugacy classes of elements of G; (Cannon &

Souvignier, 1997)

Eamonn O’Brien Matrix group recognition: status and future?

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Examples of algorithms using Soluble Radical model

◮ Determine conjugacy classes of elements of G; (Cannon &

Souvignier, 1997)

◮ Determine maximal subgroups of G; (Cannon & Holt, 2004)

and (Eick & Hulpke, 2001)

Eamonn O’Brien Matrix group recognition: status and future?

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Examples of algorithms using Soluble Radical model

◮ Determine conjugacy classes of elements of G; (Cannon &

Souvignier, 1997)

◮ Determine maximal subgroups of G; (Cannon & Holt, 2004)

and (Eick & Hulpke, 2001)

◮ Determine the automorphism group of G; (Cannon & Holt,

2003)

Eamonn O’Brien Matrix group recognition: status and future?

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Examples of algorithms using Soluble Radical model

◮ Determine conjugacy classes of elements of G; (Cannon &

Souvignier, 1997)

◮ Determine maximal subgroups of G; (Cannon & Holt, 2004)

and (Eick & Hulpke, 2001)

◮ Determine the automorphism group of G; (Cannon & Holt,

2003)

◮ Determine conjugacy classes of subgroups of G; (Cannon, Cox

& Holt, 2001)

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge I: Conjugacy classes of elements

Wall (1963), Liebeck & Seitz (2014): description of conjugacy classes and centralisers of elements of classical groups.

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge I: Conjugacy classes of elements

Wall (1963), Liebeck & Seitz (2014): description of conjugacy classes and centralisers of elements of classical groups. Gonshaw, Liebeck, O’Brien (2017): explicit listing of conjugacy classes of unipotent elements.

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge I: Conjugacy classes of elements

Wall (1963), Liebeck & Seitz (2014): description of conjugacy classes and centralisers of elements of classical groups. Gonshaw, Liebeck, O’Brien (2017): explicit listing of conjugacy classes of unipotent elements. Liebeck, O’Brien (ongoing): centralisers and conjugacy testing of unipotent elements. Centraliser of unipotent element lies in a certain parabolic; compute the unipotent radical of centraliser; write down semisimple generators.

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge I: Conjugacy classes of elements

Wall (1963), Liebeck & Seitz (2014): description of conjugacy classes and centralisers of elements of classical groups. Gonshaw, Liebeck, O’Brien (2017): explicit listing of conjugacy classes of unipotent elements. Liebeck, O’Brien (ongoing): centralisers and conjugacy testing of unipotent elements. Centraliser of unipotent element lies in a certain parabolic; compute the unipotent radical of centraliser; write down semisimple generators. Giovanni De Franceschi (2018): algorithms, which given d and q, constructs explicitly classes, centralisers, and solves conjugacy testing for classical SX(d, q). Uses our work for unipotent case.

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge I: Conjugacy classes of elements

Wall (1963), Liebeck & Seitz (2014): description of conjugacy classes and centralisers of elements of classical groups. Gonshaw, Liebeck, O’Brien (2017): explicit listing of conjugacy classes of unipotent elements. Liebeck, O’Brien (ongoing): centralisers and conjugacy testing of unipotent elements. Centraliser of unipotent element lies in a certain parabolic; compute the unipotent radical of centraliser; write down semisimple generators. Giovanni De Franceschi (2018): algorithms, which given d and q, constructs explicitly classes, centralisers, and solves conjugacy testing for classical SX(d, q). Uses our work for unipotent case. Constructive recognition: allows us to answer these questions in arbitrary repn of the classical group.

Eamonn O’Brien Matrix group recognition: status and future?

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Embed TF-group H := G/L in direct product of Aut(Ti) ≀ Sym(di), where Ti occurs di times as socle factor.

Eamonn O’Brien Matrix group recognition: status and future?

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Embed TF-group H := G/L in direct product of Aut(Ti) ≀ Sym(di), where Ti occurs di times as socle factor. Conjugacy class representatives in wreath products described theoretically (Hulpke 2000; Cannon & Holt, 2006).

Eamonn O’Brien Matrix group recognition: status and future?

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Embed TF-group H := G/L in direct product of Aut(Ti) ≀ Sym(di), where Ti occurs di times as socle factor. Conjugacy class representatives in wreath products described theoretically (Hulpke 2000; Cannon & Holt, 2006). Cannon & Souvignier (1997); Hulpke (2014); Holt: algorithms use soluble radical model to solve conjugacy within finite groups. Same tasks for exceptional groups?

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge II: Normaliser of H ≤ G

General approach: consider combinatorial or geometric invariants of

• H. Now NG(H) must preserve set of such invariants. Replace G by

stabiliser S of this invariant set and determine NS(H). Use backtrack search.

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge II: Normaliser of H ≤ G

General approach: consider combinatorial or geometric invariants of

• H. Now NG(H) must preserve set of such invariants. Replace G by

stabiliser S of this invariant set and determine NS(H). Use backtrack search. Butler (1983): NSn(H) permutes the orbital graphs of H. Theissen (1997): many refinements. Leon (1997): general backtrack procedures to construct NSn(H).

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge II: Normaliser of H ≤ G

General approach: consider combinatorial or geometric invariants of

• H. Now NG(H) must preserve set of such invariants. Replace G by

stabiliser S of this invariant set and determine NS(H). Use backtrack search. Butler (1983): NSn(H) permutes the orbital graphs of H. Theissen (1997): many refinements. Leon (1997): general backtrack procedures to construct NSn(H). Holt (1991): elements of NG(H) induce automorphisms of H. Hulpke (2008): reduces to centralisers and element conjugacy in G and calculations in Aut(H).

Eamonn O’Brien Matrix group recognition: status and future?

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Challenge II: Normaliser of H ≤ G

General approach: consider combinatorial or geometric invariants of

• H. Now NG(H) must preserve set of such invariants. Replace G by

stabiliser S of this invariant set and determine NS(H). Use backtrack search. Butler (1983): NSn(H) permutes the orbital graphs of H. Theissen (1997): many refinements. Leon (1997): general backtrack procedures to construct NSn(H). Holt (1991): elements of NG(H) induce automorphisms of H. Hulpke (2008): reduces to centralisers and element conjugacy in G and calculations in Aut(H). Coutts (2011): Specialise to G = GL(d, q). For each class Ci, for i ∈ {1, 2, 3, 5, 8}, construct overgroup M for the normaliser in G of a group H ∈ Ci. Now solve problem in M.

Eamonn O’Brien Matrix group recognition: status and future?

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Other challenges

◮ Subgroup conjugacy ◮ Maximal subgroups ◮ Intersection ◮ Stabiliser of subspaces

Eamonn O’Brien Matrix group recognition: status and future?