On finite groups isospectral to simple groups Andrey Vasil ev - PowerPoint PPT Presentation

On finite groups isospectral to simple groups Andrey Vasil ′ ev Sobolev Institute of Mathematics Group Theory Conference in Honour of Victor Mazurov Novosibirsk, 2013 1 / 23

G is a finite group | G | is the order of G π ( G ) is the set of prime divisors of | G | ω ( G ) is the spectrum of G , that is the set of its element orders G and H are isospectral if ω ( G ) = ω ( H ) Mazurov’s Conjecture Generally, if L is a finite nonabelian simple group and G is a finite group isospectral to L , then L � G � Aut( L ). Remark If V is a minimal normal abelian subgroup of a finite group H , then ω ( V ⋋ H ) = ω ( H ). So if V is nontrivial, then there are infinitely many finite groups isospectral to H . 2 / 23

Sporadic and Alternating Groups G is an arbitrary finite group. Theorem 1 (1998) Let L be a sporadic simple group and ω ( G ) = ω ( L ). If L � = J 2 , then G ≃ L . If L = J 2 , then ω ( L ) = ω ( V ⋋ A 8 ), where V ≃ 2 6 . Theorem 2 (2012) Let L be an alternating group A n , n � 5, and ω ( G ) = ω ( L ). If n �∈ { 6 , 10 } , then G ≃ L . If n = 6, then ω ( L ) = ω ( V ⋋ A 5 ), where V ≃ 2 4 . If n = 10, then ω ( L ) = ω ( V ⋋ H ), where V is abelian and H contains a section isomorphic to A 5 . 3 / 23

Exceptional Groups of Lie Type Theorem 3 (2013) If L is a simple exceptional group of Lie type, and ω ( G ) = ω ( L ), then L � G � Aut( L ). Conjecture (Question 16.24 in Kourovka Notebook ) If L is a simple exceptional group of Lie type and ω ( G ) = ω ( L ), then G ≃ L . The conjecture is proved for 2 B 2 ( q ), 2 G 2 ( q ), 2 F 4 ( q ), G 2 ( q ), E 8 ( q ) still open for 3 D 4 ( q ), F 4 ( q ), E 6 ( q ), 2 E 6 ( q ), E 7 ( q ). 4 / 23

Classical Groups Theorem 4 (2013) If L is a classical group of sufficiently large dimension and ω ( G ) = ω ( L ), then L � G � Aut( L ). L ∈ { L n ( q ) , U n ( q ) , S 2 n ( q ) , O 2 n +1 ( q ) , O ± 2 n ( q ) } ”Sufficiently large” in Th’m 4 means n � 45 for L = L n ( q ) and U n ( q ) n � 28 for L = S 2 n ( q ) and O 2 n +1 ( q ) n � 31 for O + 2 n ( q ) n � 30 for O − 2 n ( q ). Main Result Mazurov’s conjecture is valid for almost all sporadic and alternating groups, for all exceptional groups, and for classical groups of almost all dimensions. 5 / 23

Classical Groups of Smaller Dimensions Conjecture Let L be one of the following groups: L n ( q ), n � 5 U n ( q ), n � 5, and ( n , q ) � = (5 , 2) S 2 n ( q ), n � 3, and ( n , q ) �∈ { (3 , 2) , (4 , 2) } O 2 n +1 ( q ), q odd, n � 3, and ( n , q ) � = (3 , 3) 2 n ( q ), n � 5. O ε If G is a finite group with ω ( G ) = ω ( L ), then L � G � Aut L . Remark. For every group L in { L 2 ( q ) , L 3 ( q ) , U 3 ( q ) , S 4 ( q ) } we know if Mazurov’s conjecture is valid or not. 6 / 23

G. Higman (1957) Finite groups in which every element has prime power order G is EPPO-group if n is a prime power for every n ∈ ω ( G ). Theorem If G is a soluble EPPO-group, then its Fitting subgroup F ( G ) is a p -group for some prime p and G / F ( G ) is either a cyclic q -group, or a generalized quaternion group, or a group of order p a q b with cyclic Sylow subgroups. In particular, | π ( G ) | � 2. Theorem Let G be a insoluble EPPO-group. Then G has the normal series G � N � P � 1, where (i) P is the soluble radical of G and is a p -group for some prime p ; (ii) N / P is the unique normal subgroup of G / P and is a nonabelian simple group; (iii) G / N is a p -group for the same p and is cyclic or generalized quaternion. In particular, G has the unique nonabelian composition factor. 7 / 23

M. Suzuki (1962), On a class of doubly transitive groups Theorem If L is a nonabelian simple EPPO-group, then L is one of the groups: L 2 ( q ) for q = 5 , 7 , 8 , 9 and 17, L 3 (4), Sz ( q ) for q = 8 , 32. R. Brandl (1981) If G is an insoluble EPPO-group, then G is either from Suzuki’s list, or G has a nontrivial normal 2-subgroup P and G / P is one of the groups: L 2 ( q ) for q = 5 , 8 and Sz ( q ) for q = 8 , 32. 8 / 23

W. Shi If L is a simple group, ω ( L ) = ω ( G ), then G ≃ L in the following cases: L = A 5 , L 2 (7) (1984) L = L 2 (2 α ) (1987) L = Sz (2 2 m +1 ) (1992) Brandl and Shi Let L be a simple group, ω ( L ) = ω ( G ). If L = A 6 , then ω ( L ) = ω ( V ⋋ A 5 ) (1991) If L = 2 G 2 (3 2 m +1 ), then G ≃ L (1992) If L = L 2 ( q ), q � = 9, then G ≃ L (1994). Remark. A 5 ≃ L 2 (4) ≃ L 2 (5) and A 6 ≃ L 2 (9). 9 / 23

V. Mazurov, 1994 If L = L 3 (5) and ω ( G ) = ω ( L ), then G is either L or Aut( L ), an extension of L by its graph automorphism of order 2. h ( G ) is the number of pairwise non-isomorphic finite groups isospectral to G . Mazurov’s result was the first example of a group L with h ( L ) = 2. A. Zavarnitsine, 2006 If L = L 3 ( q ), q = p α ≡ 1 (mod 3), p odd, and ω ( G ) = ω ( L ), then G = L � ρ 3 i � , where 0 � i < k , 3 k || α , and ρ is a field automorphism of L of order 3 k . In particular, h ( L ) = k . Remark. Generally, in case of classical groups L we may expect 1 < h ( L ) < ∞ rather than h ( L ) = 1. 10 / 23

Theorem (... Mazurov and Shi, 1998) Let L be a sporadic simple group and ω ( G ) = ω ( L ). If L � = J 2 , then G ≃ L . If L = J 2 , then ω ( L ) = ω ( V ⋋ A 8 ), where V ≃ 2 6 . Theorem (Shi, 1987, and Mazurov, 1998) If G has a nontrivial normal abelian subgroup, then h ( G ) = ∞ . (Mazurov and Shi, 2012) If h ( G ) = ∞ , then there is a group H , isospectral to G , with a nontrivial normal abelian subgroup. 11 / 23

Let L be an alternating group A n , n � 5. Mazurov and Zavarnitsine, 1999 If a finite group G contains a nontrivial normal subgroup K and G / K ≃ L , then ω ( G ) � = ω ( L ). A. Kondratiev, Mazurov, and Zavarnitsine, 2000 Let r be a prime, r > 5. If n ∈ { r , r + 1 , r + 2 } or n = 16, and ω ( G ) = ω ( L ), then G ≃ L . I. Vakula, 2010 Let n � 21. If ω ( G ) = ω ( L ) and r is the greatest prime � n , then among composition factors of G there is a factor S ≃ A m , where r � m � n . I. Gorshkov, 2012 Let ω ( G ) = ω ( L ). If n �∈ { 6 , 10 } , then G ≃ L . 12 / 23

Let L be a simple group of Lie type over a field of order q = p α . G is a finite group with ω ( G ) = ω ( L ). Williams (+ Gruenberg-Kegel’s theorem), 1981, Kondratiev, 1989, Vasil ′ ev, Vasil ′ ev-Vdovin, 2005 Proposition If L differs from L 3 (3), U 3 (3), S 4 (3), then G has exactly one nonabelian composition factor. S ≃ Inn S � G / K � Aut( S ) K is the soluble radical of G and S is a nonabelian simple group 13 / 23

L is a nonabelian simple group, G is a group with ω ( G ) = ω ( L ) S � G = G / K � Aut S (A) G /S ? (Q) L S (C) 1 K 14 / 23

G is a cover of H , if G contains a normal subgroup K such that G / K ≃ H . H is recognizable among its covers, if ω ( H ) � = ω ( G ) for every proper cover G of H . Remark. If a simple group L is not recognizable among its covers, then h ( L ) = ∞ . Mazurov and Zavarnitsine, 2007 Let L = L n ( q ), q = p α . If n > max { 18 , p + 1 } , then L is recognizable among its covers. Maria Grechkoseeva, talk at this conference On element orders in covers of finite simple groups of Lie type Theorem (... Grechkoseeva, 2013) Let L be a finite nonabelian simple group. If L is a classical group, then suppose, in addition, that the dimension of L is at least 15. If G is a proper cover of L , then ω ( G ) � = ω ( L ). 15 / 23

ω ( L ) = ω ( G ), S is nonabelian simple, and S � G / K � Aut S . We wish to show S ≃ L ( L is quasirecognizable). GK ( G ) is the prime graph of G with the vertex set π ( G ) and r ∼ s ⇔ rs ∈ ω ( G ) ω ( G ) ⇒ GK ( G ) s ( G ) is the number of connected components of GK ( G ). Gruenberg–Kegel’s theorem: if s ( G ) > 1, then 1 either G is Frobenius, or 2-Frobenius, and s ( G ) = 2; 2 or S � G / K � Aut S and s ( S ) � s ( G ). Williams’81, Kondratiev’89 gave a description of connected components of prime graphs for all simple groups L with s ( L ) > 1. These results provided tools to prove a quasirecognizability of simple groups with disconnected prime graph. If L is an exceptional group of Lie type, then s ( L ) > 1, excepting L = E 7 ( q ) , q > 3. 16 / 23

L is a simple exceptional group of Lie type, ω ( G ) = ω ( L ) S � G / K � Aut S L S ≃ L K = 1 G ≃ L 2 B 2 (2 2 m +1 ) Shi’92 2 G 2 (3 2 m +1 ) Brandl-Shi’93 2 F 4 (2 2 m +1 ) Deng-Shi’99 E 8 ( q ) AK’02 Kondratiev’10 G 2 (4) Mazurov’02 G 2 (3 m ) Vasil ′ ev’02 G 2 ( q ), q � = 3 m , 4 V.-Staroletov’12 F 4 (2 m ) AK’03 CGMSV’04 F 4 ( p m ), p � = 2 AK’05 Gr’13 ? 3 D 4 ( q ) AK’06 Gr’13 ? E 6 ( q ), 2 E 6 ( q ) Kon’06 Gr’13 ? E 7 ( q ), q = 2 , 3 AK’05 Gr’13 ? E 7 ( q ), q > 3 VS’13 Gr’13 ? It follows that L � G � Aut( L ). 17 / 23