Effective algorithms for groups of Lie type Eamonn OBrien - PowerPoint PPT Presentation

Effective algorithms for groups of Lie type Eamonn O’Brien University of Auckland February 2015 artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Overview of lecture G “large" finite group described by generating set X . e.g. G = � X � ≤ GL ( d , q ) or G = � X � ≤ Sym ( n ) . artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Overview of lecture G “large" finite group described by generating set X . e.g. G = � X � ≤ GL ( d , q ) or G = � X � ≤ Sym ( n ) . Can we answer the following? ◮ Conjugacy classes of elements or subgroups of G ◮ Sylow p -subgroups of G ◮ Maximal subgroups of G ◮ Automorphism group of G artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Overview of lecture G “large" finite group described by generating set X . e.g. G = � X � ≤ GL ( d , q ) or G = � X � ≤ Sym ( n ) . Can we answer the following? ◮ Conjugacy classes of elements or subgroups of G ◮ Sylow p -subgroups of G ◮ Maximal subgroups of G ◮ Automorphism group of G Soluble Radical model of computation : uniform approach. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Overview of lecture G “large" finite group described by generating set X . e.g. G = � X � ≤ GL ( d , q ) or G = � X � ≤ Sym ( n ) . Can we answer the following? ◮ Conjugacy classes of elements or subgroups of G ◮ Sylow p -subgroups of G ◮ Maximal subgroups of G ◮ Automorphism group of G Soluble Radical model of computation : uniform approach. ◮ Explain the model. ◮ Discuss how to construct the model. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure G has characteristic series C of subgroups: 1 ≤ O ∞ ( G ) ≤ S ∗ ( G ) ≤ P ( G ) ≤ G artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure 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 artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure 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 ) ) = T 1 × . . . × T k where T i non-abelian simple artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure 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 ) ) = T 1 × . . . × T k where T i non-abelian simple φ : G �− → Sym ( k ) is repn of G induced by conjugation on { T 1 , . . . , T k } and P ( G ) = ker φ artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure 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 ) ) = T 1 × . . . × T k where T i non-abelian simple φ : G �− → Sym ( k ) is repn of G induced by conjugation on { T 1 , . . . , T k } and P ( G ) = ker φ P ( G ) / S ∗ ( G ) ≤ Out ( T 1 ) × . . . × Out ( T k ) and so is soluble artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Characteristic structure 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 ) ) = T 1 × . . . × T k where T i non-abelian simple φ : G �− → Sym ( k ) is repn of G induced by conjugation on { T 1 , . . . , T k } and P ( G ) = ker φ P ( G ) / S ∗ ( G ) ≤ Out ( T 1 ) × . . . × Out ( T k ) and so is soluble G / P ( G ) ≤ Sym ( k ) artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Exploiting the characteristic series C Cannon, Holt et al. (1997– ): use C in practical algorithms. 1 ≤ L := O ∞ ( G ) ≤ S ∗ ( G ) ≤ P ( G ) ≤ G artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

Exploiting the characteristic series C Cannon, Holt et al. (1997– ): use C in practical algorithms. 1 ≤ L := O ∞ ( G ) ≤ S ∗ ( G ) ≤ P ( G ) ≤ G Also compute series 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ⊳ G where N i � G and N i / N i − 1 is elementary abelian. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. Given a problem : artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. Given a problem : Solve problem first in G / L = G / N r , and then, successively, solve it in G / N i , for i = r − 1 , . . . , 0. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. Given a problem : Solve problem first in G / L = G / N r , and then, successively, solve it in G / N i , 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 T i and these are permuted under conjugation by H . artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. Given a problem : Solve problem first in G / L = G / N r , and then, successively, solve it in G / N i , 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 T i and these are permuted under conjugation by H . Problem may have nice solution for H . artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

The Soluble Radical model 1 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ≤ S ∗ ( G ) ≤ P ( G ) ≤ G where N i � G and N i / N i − 1 is elementary abelian. Given a problem : Solve problem first in G / L = G / N r , and then, successively, solve it in G / N i , 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 T i 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. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

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) artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

How do we construct the characteristic chain? Basic approach: Schreier-Sims techniques, developed first in permutation group context. Sims (1970, 1971): base and strong generating set (BSGS). Determines chain of stabilisers. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

How do we construct the characteristic chain? Basic approach: Schreier-Sims techniques, developed first in permutation group context. Sims (1970, 1971): base and strong generating set (BSGS). Determines chain of stabilisers. G ≤ GL ( d , F ) acts faithfully on V = F d ; v · g , for v ∈ V artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

How do we construct the characteristic chain? Basic approach: Schreier-Sims techniques, developed first in permutation group context. Sims (1970, 1971): base and strong generating set (BSGS). Determines chain of stabilisers. G ≤ GL ( d , F ) acts faithfully on V = F d ; v · g , for v ∈ V Now compute BSGS for G , viewed as permutation group on the vectors with base points e.g. standard basis vectors for V . artlogo Eamonn O’Brien Effective algorithms for groups of Lie type

How do we construct the characteristic chain? Basic approach: Schreier-Sims techniques, developed first in permutation group context. Sims (1970, 1971): base and strong generating set (BSGS). Determines chain of stabilisers. G ≤ GL ( d , F ) acts faithfully on V = F d ; v · g , for v ∈ V Now compute BSGS for G , viewed as permutation group on the vectors with base points e.g. standard basis vectors for V . Central problem: good subgroup chain may not exist. artlogo Eamonn O’Brien Effective algorithms for groups of Lie type