On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).     

Classification of maximal subgroups of odd index in finite groups with alternating socle

A classification of maximal subgroups of odd index in finite groups with alternating socle is obtained.     

Classification of maximal subgroups of odd index in finite simple classical groups

The classification of maximal subgroups of odd index in finite simple classical groups is obtained.     

Minimal permutation representations of finite simple classical groups. Special linear,symplectic, and unitary groups

This work was supported by the Russian Foundation for Fundamental Research, grant 93-011-1501.     

On spectra of almost simple groups with symplectic or orthogonal socle

Finite groups are said to be isospectral if they have the same sets of the orders of elements. We investigate almost simple groups H with socle S, where S is a finite simple symplectic or orthogonal group over a field of odd characteristic. We prove that if H is isospectral to S, then H/S presents a 2-group. Also we give a criterion for isospectrality of H and S in the case when S is either symplectic or orthogonal of odd dimension.     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.     

A classification of subgroups of odd unitary groups

In this paper, under the Λ-stable rank condition, we discuss the classification of subgroups of odd unitary groups and get an analogue of Sandwich Theorem.     

On finite groups isospectral to simple linear and unitary groups

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

Orthogonal, symplectic and unitary representations of finite groups

Let be a field, a finite group, and a linear representation on the finite dimensional -space . The principal problems considered are:

I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms which are -invariant.

II. If is such a form, enumerate the equivalence classes of representations of into the corresponding group (orthogonal, symplectic or unitary group).

III. Determine conditions on or under which two orthogonal, symplectic or unitary representations of are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically' equivalent.

This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) -module components of their spaces are equivalent.

We assume throughout that the characteristic of does not divide .

Solutions to I and II are given when is a finite or local field, or when is a global field and the representation is ``split'. The results for III are strongest when the degrees of the absolutely irreducible representations of are odd - for example if has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that is a local or global field, when the representations are split.

     

Maximal subgroups in finite and profinite groups

We prove that if a finitely generated profinite group is not generated with positive probability by finitely many random elements, then every finite group is obtained as a quotient of an open subgroup of . The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203--220, we then
prove that a finite group has at most maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

     

Conjugate biprimitive finite groups saturated with finite simple subgroups

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L₂(pⁿ)}, {Re(3²ⁿ⁺¹)}, and {Sz(2²ⁿ⁺¹)}. It is proved thai such groups are all isomorphic to {L₂(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 224–245, March–April, 1998.     

Almost simple groups with socle ~3D_4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O'Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T≤G≤Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to 3D4(q), then T is line-transitive, where q is a power of the prime p.     

Almost simple groups with socle 3 D 4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

Solvable subgroups of finite simple groups

This article is the author's abstract of his dissertation for the degree Doctor of Physico-mathematical Sciences. The dissertation was defended on September 17, 1973, at the session of the Academic Council on the conferring of academic degrees in the mathematical sciences of the Uralian Order of the Red Banner of Labor of the A. M. Gor'kii State University. The official opponents were: Professor A. I. Shirshov, Corresponding Member of the Academy of Sciences of the USSR and Doctor of Physicomathematical Sciences; Professor A. I. Kostrikin, Doctor of Physicomathematical Sciences; Professor A. I. Starostin, Doctor of Physicomathematical Sciences; Professor L. A. Shemetkov, Doctor of Physico-mathematical Sciences. The chief institution was the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR.Translated from Matematicheskie Zametki, Vol. 16, No. 5, pp. 833–842, November, 1974.