Large element orders and the characteristic of finite simple - PowerPoint PPT Presentation

Large element orders and the characteristic of finite simple symplectic and orthogonal groups Daniel Lytkin Novosibirsk State University Groups St Andrews 3rd – 11th August 2013 Dan Lytkin (Novosibirsk) 1 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process � � 1. find the characteristic of G ch( G ) Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process � � 1. find the characteristic of G ch( G ) 2. determine the stardard name of G Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process � � 1. find the characteristic of G ch( G ) 2. determine the stardard name of G 3. produce an explicit authomorphism with a known simple group Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition Matrix group recognition Let G = � X � ≤ GL( n, q ) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process � � 1. find the characteristic of G ch( G ) 2. determine the stardard name of G 3. produce an explicit authomorphism with a known simple group In the 2009 paper by Kantor and Seress a Monte Carlo algorithm is described for finding the characteristic of Lie-type simple groups. It involves examining the orders of a random selection of group elements. Dan Lytkin (Novosibirsk) 2 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; F := set of formulae for triples ( H, m 1 ( H ) , m 2 ( H )) for all Lie-type simple groups H ; Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; F := set of formulae for triples ( H, m 1 ( H ) , m 2 ( H )) for all Lie-type simple groups H ; 1. g := random element of G , place | g | in L ; repeat up to N times; Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; F := set of formulae for triples ( H, m 1 ( H ) , m 2 ( H )) for all Lie-type simple groups H ; 1. g := random element of G , place | g | in L ; repeat up to N times; random sample of size N contains elements of orders m 1 ( G ) , m 2 ( G ) , m 3 ( G ) with probability at least 1 − ε Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; F := set of formulae for triples ( H, m 1 ( H ) , m 2 ( H )) for all Lie-type simple groups H ; 1. g := random element of G , place | g | in L ; repeat up to N times; random sample of size N contains elements of orders m 1 ( G ) , m 2 ( G ) , m 3 ( G ) with probability at least 1 − ε There is N that depends only on n and ε ! Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic Denote by m 1 ( G ) > m 2 ( G ) > . . . the largest element orders of a group G . Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp 2 n (2 k ) and Ω ± 2 n (2 k ) . If m i ( G ) = m i ( H ) for i = 1 , 2 , then one of the following holds: 1. ch( G ) = ch( H ) . 2. *few ambiguous cases* Algorithm (simplified) Input: G = � X � ≤ GL( n, q ) and an error bound ε > 0. Output: ch( G ). 0. L := ∅ ; F := set of formulae for triples ( H, m 1 ( H ) , m 2 ( H )) for all Lie-type simple groups H ; 1. g := random element of G , place | g | in L ; repeat up to N times; random sample of size N contains elements of orders m 1 ( G ) , m 2 ( G ) , m 3 ( G ) with probability at least 1 − ε There is N that depends only on n and ε ! 2. m ′ 1 , m ′ 2 , m ′ 3 := three largest numbers in L ; Dan Lytkin (Novosibirsk) 3 / 10