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Let R be a unital subring of a commutative ring S, which is a free R-module of rank m. In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups GL(n, S) and E(n, S) in G = GL(mn,R), and showed that they are equal and coincide with the set {g ∈ G:E(n, S)g ≤ GL(n, S)} = Aut(S=R) x GL(n, S). Moreover, for any proper ideal A of R, NG (E(n, S) E(mn,R,A)), = ... (NGL(mn,R/A)) (E(n,S/SA))). In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups EO(n, S), SO(n, S),O(n, S) and GO(n, S) in G are equal and coincide with the set {g ∈ G: EO(n, S)g ≤ GO(n, S)} = Aut(S/R) x GO(n, S). Similarly, the ones of subgroups Ep(n, S), Sp(n, S), and GSp(n, S) are equal and coincide with the set {g ∈ G:Ep(n, S)g ≤ GSp(n, S)} = Aut(S/R) x GSp(n, S). Moreover, for any proper ideal A of R, NG(EO(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (EO(n,S/SA))) and NG(Ep(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (Ep(n,S/SA))). When R = S, we obtain the known results of N. A. Vavilov and V. A. Petrov. [ABSTRACT FROM AUTHOR]

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then N X (G) is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic. [ABSTRACT FROM AUTHOR]

We show that the proportion of permutations g in Sn or An such that g has even order and g|g|/2 is an involution with support of cardinality at most ⌈nε⌉ is at least a constant multiple of ε. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension n in odd characteristic that have even order and power up to an involution with (-1)-eigenspace of dimension at most ⌈nε⌉ for a linear or unitary group, or 2⌈⌊n/2⌋ε⌉ for a symplectic or orthogonal group. [ABSTRACT FROM AUTHOR]

We show that inside the real quaternionic symplectic group $${\mathrm {Sp}}(p,q)$$ , every element is conjugate to its inverse. Similar result is proved for the real quaternionic orthogonal group $$\mathrm{O }^{*}(2n)$$ . We also show that no similar result is possible in the p-adic case. [ABSTRACT FROM AUTHOR]

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol). [ABSTRACT FROM AUTHOR]

A theorem is proved on the structure of the group of isometries of a Hermitian map b : V x V → W, where V and W are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given a Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection over a set of classical subgroups of GL(V ), giving rise to the first polynomial-time solution of this old problem. The approach yields new algorithmic tools for algebras with involution, which in turn have applications to other computational problems of interest. Implementations of the various algorithms in the Magma system demonstrate their practicability. [ABSTRACT FROM AUTHOR]

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

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite simple orthogonal groups in odd dimensions are defined on cosets with respect to maximal parabolic subgroups. [ABSTRACT FROM AUTHOR]

Let V be a three-dimensional vector space over a finite field. We show that any irreducible subgroup of GL( V) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V. In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups. [ABSTRACT FROM AUTHOR]

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite symplectic groups are defined on cosets with respect to maximal parabolic subgroups. [ABSTRACT FROM AUTHOR]

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7, 9]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras. [ABSTRACT FROM AUTHOR]

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. [ABSTRACT FROM AUTHOR]

Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form. [ABSTRACT FROM AUTHOR]

We give formulae relating the value Xλ ( g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL 2 case [1]. [ABSTRACT FROM AUTHOR]

We continue the description of embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections which was begun in Part I [sb1]. We deal with the embeddings in orthogonal groups. [ABSTRACT FROM AUTHOR]

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections. [ABSTRACT FROM AUTHOR]