A note on finite group 1-cohomology via semi-direct products with applications to permutation modules

2021 ◽  
Vol 15 (4) ◽  
pp. 205-214
Morton E. Harris
1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
K.W. Gruenberg ◽  
L.G. Kovács

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.


Abstract In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups. The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
J. Sivardière

Let G be a finite group, H an invariant subgroup and F the corresponding factor group. The classes of conjugated elements of G are derived from the classes of H and F. We consider simple point groups and symmorphic space groups, which are semi-direct products H^F, then double point groups and non- symmorphic space groups, which are extensions of F by H.

D. R. Taunt

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

1981 ◽  
Vol 31 (4) ◽  
pp. 464-469 ◽  
Rolf Brandl

AbstractA finite variety is a class of finite groups closed under taking subgroups, factor groups and finite direct products. To each such class there exists a sequence w1, w2,… of words such that the finite group G belongs to the class if and only if wk(G) = 1 for almost all k. As an illustration of the theory we shall present sequences of words for the finite variety of groups whose Sylow p-subgroups have class c for c = 1 and c = 2.

2011 ◽  
Vol 10 (02) ◽  
pp. 295-301

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A5, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

2009 ◽  
Vol 79 (1) ◽  
pp. 23-30 ◽  

AbstractThe minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

2011 ◽  
Vol 10 (05) ◽  
pp. 901-914 ◽  

Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Assume that G is non-abelian and generated by two elements, one of which is an involution, and that n is relatively prime to |G|. Our first main result is the determination of the strong symmetric genus of the direct product Zn ×G in terms of n, |G|, and a parameter associated with the group G. We obtain a variety of genus formulas. Finally, we apply these results to prove that for each integer g ≥ 2, there are at least four groups of strong symmetric genus g.

1997 ◽  
Vol 39 (3) ◽  
pp. 259-268 ◽  
C. M. Campbell ◽  
I. Miyamoto ◽  
E. F. Robertson ◽  
P. D. Williams

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].

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