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

2021 ◽  
Vol 15 (4) ◽  
pp. 205-214
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Direct Products ◽  
Group 1

scholarly journals Proficient presentations and direct products of finite groups

1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
Author(s):  
K.W. Gruenberg ◽  
L.G. Kovács
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Module Structure ◽  
Direct Products ◽  
The Difference ◽  
A Finite Group ◽  
Open Question

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.

scholarly journals Chief factors in Polish groups

2021 ◽  
pp. 1-58
Author(s):  
COLIN D. REID ◽  
PHILLIP R. WESOLEK ◽  
FRANÇOIS LE MAÎTRE
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Semidirect Product ◽  
Structure Theory ◽  
Direct Products ◽  
Polish Groups ◽  
Finite Group Theory ◽  
Normal Image ◽  
Chief Factors

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.

Classes d'éléments conjugués des groupes cristallographiques

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
Author(s):  
J. Sivardière
Keyword(s):  
Finite Group ◽  
Double Point ◽  
Invariant Subgroup ◽  
Factor Group ◽  
Direct Products ◽  
Simple Point ◽  
Space Groups ◽  
Point Groups ◽  
A Finite Group

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.

Finite groups having unique proper characteristic subgroups. I

1955 ◽  
Vol 51 (1) ◽  
pp. 25-36 ◽  
Author(s):  
D. R. Taunt
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Simple Groups ◽  
Characteristic Subgroup ◽  
Direct Products ◽  
A Finite Group

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.

scholarly journals Finite varieties and groups with Sylow p-subgroups of low class

1981 ◽  
Vol 31 (4) ◽  
pp. 464-469 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Direct Products ◽  
Finite Variety ◽  
Variety Of Groups ◽  
Almost All

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.

ON A THEOREM OF HUPPERT

2011 ◽  
Vol 10 (02) ◽  
pp. 295-301
Author(s):  
JIANGTAO SHI ◽  
CUI ZHANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Group 1

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.

STRICT INEQUALITIES FOR MINIMAL DEGREES OF DIRECT PRODUCTS

2009 ◽  
Vol 79 (1) ◽  
pp. 23-30 ◽  
Author(s):  
NEIL SAUNDERS
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Negative Integer ◽  
The Other ◽  
Direct Products ◽  
Counter Example ◽  
Strict Inequalities ◽  
A Finite Group

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.

THE STRONG SYMMETRIC GENUS OF DIRECT PRODUCTS

2011 ◽  
Vol 10 (05) ◽  
pp. 901-914 ◽  
Author(s):  
COY L. MAY
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Direct Product ◽  
Direct Products ◽  
A Finite Group

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.

scholarly journals The efficiency of PSL(2, p)3 and other direct products of groups

1997 ◽  
Vol 39 (3) ◽  
pp. 259-268 ◽  
Author(s):  
C. M. Campbell ◽  
I. Miyamoto ◽  
E. F. Robertson ◽  
P. D. Williams
Keyword(s):  
Finite Group ◽  
Direct Products ◽  
Schur Multiplier ◽  
Converse Problem ◽  
Number Of Generators ◽  
Products Of Groups ◽  
A Finite Group ◽  
Finite Presentations ◽  
Minimum Deficiency

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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