On finite groups isospectral to simple symplectic and orthogonal groups

The spectrum of a finite group is the set of its element orders. Two groups are said to be isospectral if their spectra coincide. We deal with the class of finite groups isospectral to simple and orthogonal groups over a field of an arbitrary positive characteristic p. It is known that a group of this class has a unique nonabelian composition factor. We prove that this factor cannot be isomorphic to an alternating or sporadic group. We also consider the case where this factor is isomorphic to a group of Lie type over a field of the same characteristic p.     

Biplanes with flag-transitive automorphism groups of almost simple type,with classical socle

In this paper we prove that if a biplane D admits a flag-transitive automorphism group G of almost simple type with classical socle, then D is either the unique (11,5,2) or the unique (7,4,2) biplane, and G≤PSL ₂(11) or PSL ₂(7), respectively. Here if X is the socle of G (that is, the product of all its minimal normal subgroups), then X⊴G≤Aut G and X is a simple classical group.     

Maximal subgroups of odd index in finite groups with simple linear,unitary, or symplectic socle

We give a classification of maximal subgroups of odd index in finite groups whose socle is isomorphic to one of the groups PSL _n (q), PSU _n (q), or PSp _n (q) for n ≥ 13.     

Biplanes with flag-transitive automorphism groups of almost simple type,with exceptional socle of Lie type

In this paper we prove that there is no biplane admitting a flag-transitive automorphism group of almost simple type, with exceptional socle of Lie type. A biplane is a (v,k,2)-symmetric design, and a flag is an incident point-block pair. A group G is almost simple with socle X if X is the product of all the minimal normal subgroups of G, and X⊴G≤Aut (G). Throughout this work we use the classification of finite simple groups, as well as results from P.B. Kleidman’s Ph.D. thesis which have not been published elsewhere.     

On the maximality of the orthogonal groups in the symplectic groups in characteristic two

Sans résumé     

q-bosonization of the orthogonal and symplectic quantum groups

On the basis of the Gauss decomposition, the q-bosonization procedure is developed for the othogonal and symplectic quantum groups. Bibliogrphy: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 166–172. Translated by E. V. Damaskinskii.     

Homological stabilization for the orthogonal and symplectic groups

It is proved that for a commutative ring R with identity and without finite residue R, fields, the integral groups of homologies Hp(sp_2n(R)) and Hp(O_2n(R)) for a fixed p do not vary with the growth of n only if n⩾2p+dim X Here dim X is the Krull dim X-dimension of the spectrum of the maximal ideals of the ring R. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 222–228, 1987.     

Pieri algebras for the orthogonal and symplectic groups

We study the structure of a family of algebras which encodes an iterated version of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to a Hibi algebra. There is also a parallel theory for the complex symplectic group.     

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.     

2-signalizers in almost simple groups

This paper offers an exhaustive study of certain 2-signalizers in known non-sporadic finite simple groups. The main result of this paper is relevant to the Generation-2 proof of the Classification of Finite Simple Groups.     

q-Deformed character theory for infinite-dimensional symplectic and orthogonal groups

Selecta Mathematica - We show that every finitely generated conical refinement monoid can be represented as the monoid $$mathcal V (R)$$ of isomorphism classes of finitely generated projective...     

On factorization of finite almost simple groups by nonconjugate maximal subgroups

We answer in the negative Question 10.34 by Monakhov in the Kourovka Notebook in the class of almost simple groups.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.