Extraspecial Group Research Articles

On the cohomology of the sporadic simple group J4

In this paper we calculate part of the integral cohomology ring of the sporadic simple group J4; this group has order 221.33.5.7. 113.23.29.31.37.43. More precisely, we obtain all of the cohomology ring except for the 2-primary part. As the cohomology has already been written down [9] at the primes which divide the group order only once, we concentrate here on the primes 3 and 11. In both of these cases the Sylow p-subgroups are extraspecial of order p3 and exponent p. We use the method which identifies the p-primary cohomology with the ring of stable classes in the cohomology of a Sylow p-subgroup. The stable classes are all invariant under the action of the Sylow p-normalizer; and some time is spent finding invariant classes in the cohomology ring of , the extraspecial group. Section 2 studies the prime 11: the invariant classes are the stable classes, because the Sylow 11-subgroups have the Trivial Intersection (T.I.) property. In Section 3 we study the prime 3, and see that all conditions for invariant classes to be stable reduce to one condition.

Extraspecial towers and weil representations

This paper was motivated by a remarkable group, the maximal subgroup $M=S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}\ltimes2^{6+1}_{-}$ of the sporadic simple group ${\rm Fi}_{23}$, where $S_3$ is the symmetric group of degree 3, and $2^{2+1}_{-}$, $3^{2+1}$ and $2^{6+1}_{-}$ denote extraspecial groups. The representation $3^{2+1}\to{\rm GL}(3,\mathbb{F}_4)\to{\rm GL}(6,\mathbb{F}_2)$ extends (remarkably) to $S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}$ and preserves a quadratic form (of minus type) which allows the construction of $M$. The paper describes certain (Weil) representations of extraspecial groups which extend, and preserve various forms. Incidentally, $M$ is a remarkable solvable group with derived length 10, and composition length 24.

The Cohomology of Extraspecial Groups

The Cohomology of Extraspecial Groups

Spectra of p-elements in the normalizer of an extraspecial linear group

Spectra of p-elements in the normalizer of an extraspecial linear group

A theorem of Hall-Higman type for groups of odd order

In Theorem B of [1], a very general Theorem of Hall-Higman type for groups of odd order was obtained. We mentioned there the possibility to drop the hypothesis (]A], r - 1) = 1 in the statement. The purpose of this note is to do it so. We give also an example showing that the assumption of r being odd is necessary. Our proof will consist into an extension of Theorem A of [1] to the case in which a group of odd order acts on an extraspecial r-group not necessarily centralizing the centre. By Theorem A of [1], we may suppose that r _-> 7. We use some auxiliary results of [l], too. We begin with two preliminary lemmas: Lemma 1. Let G be a group of odd order such that every Sylow subgroup of the Fitting subgroup F(G) of G is the central product of a cyclic group and an extraspecial group of prime exponent. Suppose that V is a KG-module, K a field, and [V,y] = V /f 1 ~ y ~ Z(F(G)). If x is a nontrivial element of G, then

Spinor groups and algebraic coding theory

The purpose of this paper is to explore the equivalence between the abelian subgroups of Ṽ(n), the “diagonal” extra-special 2-group of the compact, simple, simply-connected Lie group Spin( n), and the self-orthogonal linear binary codes of algebraic coding theory. In particular, the basic abstract structure theory of the abelian subgroups of Ṽ(n) is reflected in the distinction between ordinary self-orthogonal codes and even self-orthogonal codes. Work of Quillen on the equivariant cohomology of Spin( n) affects the classification of even self-orthogonal codes.

Dual modules and group actions on extra-special groups

Dual modules and group actions on extra-special groups

The Number of Trace-Valued Forms and Extraspecial Groups

The Number of Trace-Valued Forms and Extraspecial Groups

Gruppi semiextraspeciali di esponentep

Semi-extraspecial groups are defined by B. Beisiegel in [1] as a generalization of extraspecial groups. In this paper we characterize these groups in some different ways and we prove several their properties. Connections between these groups and other algebraic and geometrical structures are investigated, and two important subclasses are characterized in geometrical terms.

On the integral cohomology of extraspecial 2-groups

On the integral cohomology of extraspecial 2-groups

Nilpotent by Supersolvable M-Groups

A character of a finite group G is monomial if it is induced from a linear (degree one) character of a subgroup of G. A group G is an M-group if all its complex irreducible characters (the set Irr(G)) are monomial.In [1], Dade gave an example of an M-group with a normal subgroup which is itself not an M-group. In his group G, the supersolvable residual N is an extra special 2-group and G/N is supersolvable of even order. Moreover, the prime 2 is used in such a way that no analogous construction is possible in the case that |N| or |G:N| is odd. This led Isaacs in [8] and Dade in [2] to consider the effect of certain “oddness“ hypotheses in the study of monomial characters.Our main results are in the same spirit. Although our techniques seem to require a restrictive assumption on the supersolvable residual of the groups we consider, our theorems provide more evidence that under fairly general circumstances normal subgroups of M-groups should be M-groups.

Orthogonal Geometries over GF(2) and actions of Extra-special 2-Groups on Translation Planes

Orthogonal Geometries over GF(2) and actions of Extra-special 2-Groups on Translation Planes

On Thompson's simple group

Recently, Thompson [9] constructed a new simple group E of order 2 15 ∘ 3 105 37 2 ∘ 13 ∘ 19 ∘ 31 = 90, 745, 943, 887, 872, 000. In particular, E contains only one conjugacy class of involutions, and if e is an involution in E then C E ( e) is a (nonsplit) extension of an extra-special 2-group of order 2 9 by A 9, the alternating group of degree 9. The aim of this paper is to prove the following result.

A 2-local characterization of the simple group E

The purpose of this paper is to characterize the simple group E, [6] by a particular 2-local subgroup. First, some properties of a Sylow 2-subgroup of the 2-local subgroup will be studied. Second, we will show that a Sylow 2-subgroup of the 2-local subgroup is a Sylow 2-subgroup of the simple group in question. Finally, we will exhibit the centralizer of an involution and quote a result of Parrott [5] that characterizes the group E by the centralizer of an involution. The 2-local subgroup is the normalizer of an elementary abelian subgroup V of order 32. The normalizer of V is the (unique) nonsplit extension of 17 by GL(5, 2). This extension was shown to exist by Thompson [7], and its uniqueness was shown by Dempwolff [l]. For notation, let G be a simple group containing N = :V,( V) z 25GL(5, 2) as a 2-local subgroup. The centralizer of a central involution T+ in N is a group C,(v,) such that C,-(v~)/O,(C,(V,)) _c A,, with O,(C,(v,)) isomorphic to an extra-special group of order 29 of + type. The precise statement is:

On an embedding of certainp-groups

It is shown that ap-group with cyclic centre can be embedded in a finite group as a normal subgroup contained in its Frattini subgroup if and only if it is either an extraspecial 2-group of order at least 27 or the central product of a cyclic groupQ of order ≧4 and an extraspecial groupE of order ≧25, amalgamating Ω1(Q) and the centre ofE.

Automorphisms of extra special groups and nonvanishing degree 2 cohomology

Publisher Summary This chapter discusses the automorphisms of extra-special groups and nonvanishing degree 2 cohomology. If E be an extra-special group of order 2 n+1 , then n ≥ 1. The chapter identifies (E) with the relevant orthogonal group 0 ɛ (2n, 2), ɛ = ±.

The mod 2 cohomology rings of extra-special 2-groups and the spinor groups

The mod 2 cohomology rings of extra-special 2-groups and the spinor groups