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

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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

• rders

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.

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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 = J2, then G ≃ L. If L = J2, then ω(L) = ω(V ⋋ A8), where V ≃ 26.

Theorem 2 (2012)

Let L be an alternating group An, n 5, and ω(G) = ω(L). If n ∈ {6, 10}, then G ≃ L. If n = 6, then ω(L) = ω(V ⋋ A5), where V ≃ 24. If n = 10, then ω(L) = ω(V ⋋ H), where V is abelian and H contains a section isomorphic to A5.

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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 2B2(q), 2G2(q), 2F4(q), G2(q), E8(q) still open for 3D4(q), F4(q), E6(q), 2E6(q), E7(q).

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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 ∈ {Ln(q), Un(q), S2n(q), O2n+1(q), O±

2n(q)}

”Sufficiently large” in Th’m 4 means n 45 for L = Ln(q) and Un(q) n 28 for L = S2n(q) and O2n+1(q) n 31 for O+

2n(q)

n 30 for O−

2n(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.

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Classical Groups of Smaller Dimensions

Conjecture

Let L be one of the following groups: Ln(q), n 5 Un(q), n 5, and (n, q) = (5, 2) S2n(q), n 3, and (n, q) ∈ {(3, 2), (4, 2)} O2n+1(q), q odd, n 3, and (n, q) = (3, 3) Oε

2n(q), n 5.

If G is a finite group with ω(G) = ω(L), then L G Aut L.

• Remark. For every group L in {L2(q), L3(q), U3(q), S4(q)} we

know if Mazurov’s conjecture is valid or not.

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• 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,

• r a generalized quaternion group, or a group of order paqb 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.

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• 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: L2(q) for q = 5, 7, 8, 9 and 17, L3(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: L2(q) for q = 5, 8 and Sz(q) for q = 8, 32.

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• W. Shi

If L is a simple group, ω(L) = ω(G), then G ≃ L in the following cases: L = A5, L2(7) (1984) L = L2(2α) (1987) L = Sz(22m+1) (1992)

Brandl and Shi

Let L be a simple group, ω(L) = ω(G). If L = A6, then ω(L) = ω(V ⋋ A5) (1991) If L = 2G2(32m+1), then G ≃ L (1992) If L = L2(q), q = 9, then G ≃ L (1994).

• Remark. A5 ≃ L2(4) ≃ L2(5) and A6 ≃ L2(9).

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• V. Mazurov, 1994

If L = L3(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 = L3(q), q = pα ≡ 1 (mod 3), p odd, and ω(G) = ω(L), then G = Lρ3i, where 0 i < k , 3k||α, and ρ is a field automorphism of L of order 3k. In particular, h(L) = k.

• Remark. Generally, in case of classical groups L we may expect

1 < h(L) < ∞ rather than h(L) = 1.

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Theorem (... Mazurov and Shi, 1998)

Let L be a sporadic simple group and ω(G) = ω(L). If L = J2, then G ≃ L. If L = J2, then ω(L) = ω(V ⋋ A8), where V ≃ 26.

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.

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Let L be an alternating group An, 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 ≃ Am, where r m n.

• I. Gorshkov, 2012

Let ω(G) = ω(L). If n ∈ {6, 10}, then G ≃ L.

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Let L be a simple group of Lie type over a field of order q = pα. 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 L3(3), U3(3), S4(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

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L is a nonabelian simple group, G is a group with ω(G) = ω(L) S G = G/K Aut S

K S G/S (C)

1

(A)

?

(Q) L

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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 = Ln(q), q = pα. 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).

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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 = E7(q), q > 3.

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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

2B2(22m+1)

Shi’92

2G2(32m+1)

Brandl-Shi’93

2F4(22m+1)

Deng-Shi’99 E8(q) AK’02 Kondratiev’10 G2(4) Mazurov’02 G2(3m) Vasil′ev’02 G2(q), q = 3m, 4 V.-Staroletov’12 F4(2m) AK’03 CGMSV’04 F4(pm), p = 2 AK’05 Gr’13 ?

3D4(q)

AK’06 Gr’13 ? E6(q), 2E6(q) Kon’06 Gr’13 ? E7(q), q = 2, 3 AK’05 Gr’13 ? E7(q), q > 3 VS’13 Gr’13 ? It follows that L G Aut(L).

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If L is a classical group, then its prime graph is generally connected. a coclique is any subset of pairwise non-adjacent vertices in a graph t(G) is a maximal size of a coclique in GK(G) t(r, G) is a maximal size of a coclique, containing r ∈ π(G). Vasil′ev (2005): if t(G) 3 and t(2, G) 2, then

1 S G/K Aut S; 2 t(S) t(G) − 1; 3 t(2, S) = t(2, G) or S ≃ L2(q).

V.-Vdovin’2005 gave an adjacency criterion in prime graphs of all finite simple groups. These results allowed to attack a general case of Mazurov’s conjecture for classical groups. They were supported by an arithmetical description of spectra of classical groups (A. Buturlakin, 2007-2010) and V.-Vdovin’11 description of cocliques in prime graphs of simple groups.

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Theorem

Let L be a simple classical group over a field of characteristic p, and L ∈ {L3(3), U3(3), S4(3)}. Suppose G is a finite group with ω(G) = ω(L). Then G contains the unique nonabelian composition factor S and one of the following holds S ≃ L L ∈ {L2(9), U3(5), U5(2)}. L is a symplectic or orthogonal group of dimension at most 17. S is a group of Lie type over a field of characteristic v = p. Grechkoseeva, Vasil′ev, and Staroletov, 2011 (linear and unitary) GV-Mazurov’2009 and GV’2013 (symplectic and orthogonal) It reduces the problem to the case when the characteristic of S is distinct from characteristic of L.

• Remark. GV-Mazurov’2009 gave the possibility to prove another

longstanding conjecture.

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Conjecture (Shi Wujie, 1987)

Every finite simple group is uniquely determined by its order and spectrum in the class of all finite groups.

• W. Shi, J. Bi, H. Cao, M. Xu, 1987,...,2003

Shi’s conjecture is valid for all simple groups except symplectic and

• rthogonal groups (more precisely, except simple groups O+

2n(q)

with n even, O2n+1(q) and S2n(q)). Grechkoseeva, Mazurov and Vasil′ev, 2009: Shi’s conjecture is true for remaining groups. It follows

Theorem

If L is a finite simple group, and G is a finite group with |G| = |L| and ω(G) = ω(L), then G ≃ L.

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Theorem (Vasil′ev)

Let L be a simple classical group over a finite field of characteristic p, and G be a finite group with ω(G) = ω(L). Suppose that n 45 for L ∈ {Ln(q), Un(q)}, n 28 for L ∈ {S2n(q), O2n+1(q)}, n 31 for L = O+

2n(q), and n 30 for L = O− 2n(q). Then G has

the unique nonabelian composition factor S, and S is not isomorphic to a group of Lie type over a finite field of a characteristic distinct from p. The hypothesis can be reformulated: L is a classical group with t(L) 23, i.e., GK(L) contains a coclique of size at least 23.

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Sketch of the proof L is a simple classical group over Fq, q = pα, and t(L) 23. ω(G) = ω(L) and G is a counterexample. GK(G) = GK(L) ⇒ t(G) = t(L) 23, t(2, G) = t(2, L) 2. S G = G/K Aut(S), S is simple of Lie type. t(L) increases when so does a dimension n of L, e.g., t(L) = [(n + 1)/2] for L ∈ {Ln(q), Un(q)} and n 13. t(S) t(L) − 1 22, so S is a classical group.

1 Restrictions on K and G/S. 2 t(S) = t(L) and t(p, L) = t(p, S). 3 We construct a set of pairwise coprime numbers kj, (j ∈ J)

(|J| ≈ n/3) such that for every j: (i) kj ∈ ω(L), (ii) pkj ∈ ω(L), (iii) kj coprime to |K| · |G/S|. Now (i) and (iii) imply that kj ∈ ω(S) for every j ∈ J, but there is at least

• ne i ∈ J with pki ∈ ω(S) which contradicts (ii).

This proves the theorem, so the main result of the talk also holds.

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Main result

For almost all sporadic and alternating simple groups, for all exceptional simple groups, and for classical simple groups of almost all dimensions, if G is a finite group isospectral to a nonabelian simple group L, then L G Aut(L).

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