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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 v 1 > v 2 > v 3 such that a member of L has order divisible by a primitive prime divisor of one of p v i − 1. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 v 1 > v 2 > v 3 such that a member of L has order divisible by a primitive prime divisor of one of p v i − 1. Usually { v 1 , v 2 , v 3 } determines | G | and name of G . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 v 1 > v 2 > v 3 such that a member of L has order divisible by a primitive prime divisor of one of p v i − 1. Usually { v 1 , v 2 , v 3 } determines | G | and name of G . Altseimer & Borovik (2002): distinguish between PSp ( 2 m , q ) and Ω( 2 m + 1 , q ) , q odd and m ≥ 3. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Finding the characteristic BKPS and other algorithms assume that input G is a simple group of 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 ? artlogo Eamonn O’Brien Matrix group recognition: status and future?
Finding the characteristic BKPS and other algorithms assume that input G is a simple group of 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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Finding the characteristic BKPS and other algorithms assume that input G is a simple group of 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 , F r ) . Now read off r . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Finding the characteristic BKPS and other algorithms assume that input G is a simple group of 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 , F r ) . Now read off r . Kantor & Seress (2009): version for matrix groups. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Constructive recognition C = � X � ≤ GL ( d , q ) where C is (quasi)simple. C is standard copy, sometimes known as “gold copy". artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Using standard generators C = � X � artlogo
Using standard generators C = � X � � Y � = G artlogo
Using standard generators Find S = w ( X ) S C = � X � � Y � = G artlogo
Using standard generators Find S = w ( X ) S Find ¯ C = � X � S = w ( Y ) � Y � = G ¯ S artlogo
Using standard generators Find S = w ( X ) S Find ¯ C = � X � S = w ( Y ) Define φ : C �→ G : S �→ ¯ S � Y � = G ¯ S artlogo
Using standard generators Find S = w ( X ) S Find ¯ C = � X � S = w ( Y ) Define φ : C �→ G : S �→ ¯ h S h = w ( S ) � Y � = G ¯ S artlogo
Using standard generators Find S = w ( X ) S Find ¯ C = � X � S = w ( Y ) Define φ : C �→ G : S �→ ¯ h S h = w ( S ) Thus ¯ h = w ( ¯ ¯ S ) � Y � = G h ¯ S artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ( d 4 log q ) measured in field operations. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ( d 4 log q ) measured in field operations. Theorem (Liebeck & O’B, 2016) Similar statement for exceptional groups of rank ≥ 2 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ( d 4 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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ( d 4 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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Fundamental base case SL 2 ( q ) Conder, L-G, O’B (2006): defining characteristic repns relying on discrete log oracle. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Fundamental base case SL 2 ( 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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Fundamental base case SL 2 ( 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: SL 3 ( q ) , SU 3 ( q ) , Ω ǫ ( d , q ) where d ≤ 7. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Constuctive recognition algorithms Jambor et al. (2013): constructive recognition algorithms for A n and S n . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Constuctive recognition algorithms Jambor et al. (2013): constructive recognition algorithms for A n and S n . Bray, Wilson and others: standard generators and algorithms for sporadic groups. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
CompositionTree Bäärnhielm, Holt, Leedham-Green, O’B (2014): algorithm which exploits geometry and constructive recognition to construct composition series (and more) for G . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Tree is constructed in right depth-first order. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 . artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 I 1 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 H I 1 I 1 I 2 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 H H I 1 I 1 I 1 K 2 I 2 I 2 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 H H H K 1 I 1 I 1 I 1 I 1 K 2 I 2 K 2 I 2 I 2 artlogo Eamonn O’Brien Matrix group recognition: status and future?
Constructing kernels Assume φ : H �− → I where K = ker φ . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Constructing kernels Assume φ : H �− → I where K = ker φ . H K I artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 artlogo Eamonn O’Brien Matrix group recognition: status and future?
Output of CompositionTree Given G = � X � ≤ GL ( d , q ) as input. Output : ◮ a composition series: 1 = G 0 ⊳ G 1 ⊳ G 2 · · · ⊳ G m = G . ◮ A representation S k = � X k � of G k / G k − 1 ◮ Effective maps τ k : G k → S k , φ k : S k → G k τ k epimorphism with kernel G k − 1 ◮ Map to write g ∈ G as word in X . artlogo Eamonn O’Brien Matrix group recognition: status and future?
Output of CompositionTree Given G = � X � ≤ GL ( d , q ) as input. Output : ◮ a composition series: 1 = G 0 ⊳ G 1 ⊳ G 2 · · · ⊳ G m = G . ◮ A representation S k = � X k � of G k / G k − 1 ◮ Effective maps τ k : G k → S k , φ k : S k → G k τ k epimorphism with kernel G k − 1 ◮ Map to write g ∈ G as word in X . Construct presentation for group defined by tree and verify that G satisfies the relations. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Output of CompositionTree Given G = � X � ≤ GL ( d , q ) as input. Output : ◮ a composition series: 1 = G 0 ⊳ G 1 ⊳ G 2 · · · ⊳ G m = G . ◮ A representation S k = � X k � of G k / G k − 1 ◮ Effective maps τ k : G k → S k , φ k : S k → G k τ k epimorphism with kernel G k − 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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Output of CompositionTree Given G = � X � ≤ GL ( d , q ) as input. Output : ◮ a composition series: 1 = G 0 ⊳ G 1 ⊳ G 2 · · · ⊳ G m = G . ◮ A representation S k = � X k � of G k / G k − 1 ◮ Effective maps τ k : G k → S k , φ k : S k → G k τ k epimorphism with kernel G k − 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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Composition factors in polynomial-time? artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 2 G 2 ( q ) . artlogo Eamonn O’Brien Matrix group recognition: status and future?
What can we do with outcome? artlogo Eamonn O’Brien Matrix group recognition: status and future?
What can we do with outcome? Use it as infrastructure for Soluble Radical model of computation : uniform approach to computations with linear groups. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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. artlogo Eamonn O’Brien Matrix group recognition: status and future?
Characteristic structure Finite G has characteristic series C of subgroups: 1 ≤ O ∞ ( G ) ≤ S ∗ ( G ) ≤ P ( G ) ≤ G artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ) ) = T 1 × . . . × T k where T i non-abelian simple artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 ) ) = 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 Matrix group recognition: status and future?
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 ) ) = 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 Matrix group recognition: status and future?
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 ) ) = 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 Matrix group recognition: status and future?
Exploiting the characteristic series C Cannon, Holt et al. (2000s): use C in practical algorithms. 1 ≤ L := O ∞ ( G ) ≤ S ∗ ( G ) ≤ P ( G ) ≤ G artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 = N 0 ⊳ N 1 ⊳ · · · ⊳ N r = L ⊳ G where N i / N i − 1 is elementary abelian. artlogo Eamonn O’Brien Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
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 Matrix group recognition: status and future?
Examples of algorithms using Soluble Radical model ◮ Determine conjugacy classes of elements of G ; (Cannon & Souvignier, 1997) artlogo Eamonn O’Brien Matrix group recognition: status and future?
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) artlogo Eamonn O’Brien Matrix group recognition: status and future?
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) artlogo Eamonn O’Brien Matrix group recognition: status and future?
Recommend
More recommend
