scholarly journals On a class of finite groups

1975 ◽  
Vol 19 (3) ◽  
pp. 290-291
Author(s):  
J. W. Wamsley
Keyword(s):  
Finite Groups ◽  
Normal Closure ◽  
Number Of Generators ◽  
Image Position

Let G be a finite P-group. Denote dim H1 (G, Zp) by d(G) and dimH2(G, Zp) by r(G), then d(G) is the minimal number of generators of G and G has a presentation where F is free on x1, …, xd(G) and R is the normal closure in F of R1, …, Rm. We have always that m ≧ r(G) = d(R/[F, R]) and we say that G belongs to a class, Gp, of the finite pgroups if m = r(G). It is well known (see for example Johnson and Wamsley (1970)) that if G and H are finite p-groups then r(G x H) = r(G) + r(H) + d(G)d(H) and hence G, H∈Gp implies Gx H∈Gp, also it is shown in Wamsley (1972) that if G is any finite pgroup then there exists an H∈Gp such that G x H belongs to Gp. Let G1 = G and Gk = Gk-1 x G then we show in this note that if G is any finite p-group, there exists an integer n(G), such that Gk∈Gp for alal k ≧ n(G).

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

Finite groups with weakly ℋC-embedded subgroups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050093 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. For each prime [Formula: see text] dividing the order of [Formula: see text] let [Formula: see text] be a Sylow [Formula: see text]-subgroup of [Formula: see text]. We fix a subgroup of [Formula: see text] of order [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that every subgroup of [Formula: see text] of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

scholarly journals Representation of finite groups as short products of subsets

1994 ◽  
Vol 49 (3) ◽  
pp. 463-467 ◽  
Author(s):  
Xingde Jia
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Image Position

Let M be a finite quasigroup of order n. For any integer k ≥ 2, let H(k, M) be the smallest positive integer h such that there exist h subsets Ai (i = 1, 2, …, h) such that Ai … Ah = M and |Ai| = k for every i = 1, 2, …, h. Define H(k, n) = max H(k, M). It is proved in this paper that.

Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension

1996 ◽  
Vol 120 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Sanju L. Velani
Keyword(s):  
Hyperbolic Space ◽  
Finite Groups ◽  
Discrete Subgroup ◽  
Dimensional Euclidean Space ◽  
Simultaneous Diophantine Approximation ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Fundamental Polyhedron ◽  
Image Position ◽  
Geometrically Finite Groups

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

FC-nilpotent and FC-soluble groups

1956 ◽  
Vol 52 (3) ◽  
pp. 391-398 ◽  
Author(s):  
A. M. Duguid ◽  
D. H. McLain
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Abelian Groups ◽  
Characteristic Subgroup ◽  
Arbitrary Group ◽  
Soluble Groups ◽  
Image Position

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

On the Rank of a p-Group of Class 2

1983 ◽  
Vol 26 (1) ◽  
pp. 101-105 ◽  
Author(s):  
U. H. M. Webb
Keyword(s):  
Stability Number ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
The Stability ◽  
Image Position

AbstractLet d(G) denote the minimal number of generators of the finite p-group G, r(G) the maximum over all subgroups H of G of d(H) and ra(G) the maximum over all abelian subgroups H of G of d(H). If G is of class two it is clear thatBy considering properties of the stability number of graphs we construct examples which show that any value of r(G) within these bounds can occur.

Minimal Relations for Certain Wreath Products of Groups

1970 ◽  
Vol 22 (5) ◽  
pp. 1005-1009 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Wreath Products ◽  
Number Of Generators ◽  
Products Of Groups ◽  
Image Position

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:d(G) = dimKH1(G, K), the minimal number of generators of G,r(G) = dimKH2(G, K),r′(G) = the minimal number of relations required to define G,where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.

scholarly journals Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 158
Author(s):  
Li Zhang ◽  
Li-Jun Huo ◽  
Jia-Bao Liu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Normal Closure ◽  
A Finite Group ◽  
New Applications ◽  
New Criterion

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G, and H ( G ) is the set of all H -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G. As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ N G ( P ) is p-nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S-quasinormal in G, which means that H T permutes with every Sylow subgroup of G.

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