Note on the abstract group (2,3,7;9)

1966 ◽  
Vol 62 (1) ◽  
pp. 7-10 ◽  
Author(s):  
John Leech
Keyword(s):  
Finite Groups ◽  
Factor Group ◽  
Abstract Group ◽  
Image Position

The abstract groupis finite for n = 4,6,7,8, and the relations are incompatible for n = 1,2,3,5. A criterion of Coxeter ((1)) suggests that (2,3,7; n) should be infinite for all n ≥ 9, but its applicability to these groups is unproved, and it is not known whether there are any further examples of finite groups (2,3,7; n). However, (2,3,7; 9) has been proved infinite by Sims ((3)), and it follows at once that (2,3,7; n) is infinite whenever n is a multiple of 9 as it then has an infinite factor group.

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 which are c-absolute central factor group and the union of Ac-centralizers

2020 ◽  
pp. 2150042
Author(s):  
Esmat Alamshahi ◽  
Mohammad Reza R. Moghaddam ◽  
Farshid Saeedi
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Factor Group ◽  
Cyclic Groups ◽  
Central Factor ◽  
Central Factors

Let [Formula: see text] be a group and [Formula: see text] be the [Formula: see text]-absolute center of [Formula: see text], that is, the set of all elements of [Formula: see text] fixed by all class preserving automorphisms of [Formula: see text]. In this paper, we classify all finite groups [Formula: see text], whose [Formula: see text]-absolute central factors are isomorphic to the direct product of cyclic groups, [Formula: see text] and [Formula: see text]. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of [Formula: see text] for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

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.

On groups satisfying the converse of Lagrange's theorem

1974 ◽  
Vol 75 (1) ◽  
pp. 25-32 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Factor Group ◽  
Proved Theorem ◽  
Supersoluble Groups ◽  
Odd Order ◽  
Sylow Tower ◽  
Rank Of A Group ◽  
Lagrange's Theorem

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

Presentations of the Groups SL(2,m) And PSL(2,m)

1972 ◽  
Vol 24 (6) ◽  
pp. 1129-1131 ◽  
Author(s):  
J. G. Sunday
Keyword(s):  
Factor Group ◽  
Generators And Relations ◽  
Image Position

In this paper, we refine the presentations of Behr and Mennicke [1] for SL(2,m)and PSL(2,m)wheremis odd. The group SL(2,m)is first shown to be presented by the following system of generators and relations:1.1The group PSL(2,m)appears as the factor group1.2This simplification then permits us to use the results of Schur [3] to establish three-relation presentations for these groups.

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.

scholarly journals Maximal sum-free sets in abelian groups of order divisible by three: Corrigendum

1972 ◽  
Vol 7 (2) ◽  
pp. 317-318 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Arithmetic Progression ◽  
Abelian Groups ◽  
Factor Group ◽  
Image Position ◽  
Free Set ◽  
Free Sets

The last step of the proof in [2] was omitted. To complete the argument, we proceed in the following way. We had shown that H = H(S) = H(S+S) = H(S-S), that |S-S| = 2|S| - |H| and hence that in the factor group G* = G/H of order 3m, the maximal sum-free set S* = S/H and its set of differences S* - S* are aperiodic, withso thatBy (1) and Theorem 2.1 of [1], S* - S* is either quasiperiodic or in arithmetic progression.

Sign in / Sign up

Export Citation Format

Share Document