scholarly journals A note on groups with non-central norm

1994 ◽  
Vol 36 (1) ◽  
pp. 37-43 ◽  
Author(s):  
R. A. Bryce ◽  
L. J. Rylands
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Quaternion Group ◽  
Order Four

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

scholarly journals Finite Groups Which Admit An Automorphism With Few Orbits

1960 ◽  
Vol 12 ◽  
pp. 73-100 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Groups ◽  
Study Groups ◽  
Quaternion Group ◽  
Fixed Integer ◽  
Fixed Element ◽  
Odd Order ◽  
Special Case

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

scholarly journals Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

scholarly journals Criteria for Commutativity in Large Groups

1998 ◽  
Vol 41 (1) ◽  
pp. 65-70 ◽  
Author(s):  
A. Mohammadi Hassanabadi ◽  
Akbar Rhemtulla
Keyword(s):  
Abelian Group ◽  
List Type ◽  
Quaternion Group ◽  
Large Groups ◽  
Positive Integers ◽  
Finite Exponent

AbstractIn this paper we prove the following:1.Let m ≥ 2, n ≥ 1 be integers and let G be a group such that (XY)n = (YX)n for all subsets X, Y of size m in G. Thena)G is abelian or a BFC-group of finite exponent bounded by a function of m and n.b)If m ≥ n then G is abelian or |G| is bounded by a function of m and n.2.The only non-abelian group G such that (XY)2 = (YX)2 for all subsets X, Y of size 2 in G is the quaternion group of order 8.3.Let m, n be positive integers and G a group such that for all subsets Xi of size m in G. Then G is n-permutable or |G| is bounded by a function of m and n.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Certain Groups of Orthonormal Step Functions

1957 ◽  
Vol 9 ◽  
pp. 413-425 ◽  
Author(s):  
J. J. Price
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Compact Abelian Group ◽  
Walsh Functions ◽  
Cyclic Groups ◽  
Step Functions ◽  
Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

Subgroups like Wielandt's in finite soluble groups

1990 ◽  
Vol 107 (2) ◽  
pp. 239-259 ◽  
Author(s):  
R. A. Bryce
Keyword(s):  
Internal Structure ◽  
Direct Product ◽  
Periodic Group ◽  
Quaternion Group ◽  
Soluble Groups ◽  
Parent Group ◽  
Dedekind Group ◽  
Points Of View ◽  
Hamiltonian Groups

In 1935 Baer[1] introduced the concept of kern of a group as the subgroup of elements normalizing every subgroup of the group. It is of interest from three points of view: that of its structure, the nature of its embedding in the group, and the influence of its internal structure on that of the whole group. The kern is a Dedekind group because all its subgroups are normal. Its structure is therefore known exactly (Dedekind [7]): if not abelian it is a direct product of a copy of the quaternion group of order 8 and an abelian periodic group with no elements of order 4. As for the embedding of the kern, Schenkman[13] shows that it is always in the second centre of the group: see also Cooper [5], theorem 6·5·1. As an example of the influence of the structure of the kern on its parent group we cite Baer's result from [2], p. 246: among 2-groups, only Hamiltonian groups (i.e. non-abelian Dedekind groups) have nonabelian kern.

scholarly journals Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850065
Author(s):  
Alireza Abdollahi ◽  
Majid Arezoomand
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
Finite Nilpotent Group ◽  
Closed Group ◽  
Odd Order ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

Let [Formula: see text] be any group and [Formula: see text] be a subgroup of [Formula: see text] for some set [Formula: see text]. The [Formula: see text]-closure of [Formula: see text] on [Formula: see text], denoted by [Formula: see text], is by definition, [Formula: see text] The group [Formula: see text] is called [Formula: see text]-closed on [Formula: see text] if [Formula: see text]. We say that a group [Formula: see text] is a totally[Formula: see text]-closed group if [Formula: see text] for any set [Formula: see text] such that [Formula: see text]. Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

The Group of Units of the Integral Group Ring ZS3

1972 ◽  
Vol 15 (4) ◽  
pp. 529-534 ◽  
Author(s):  
I. Hughes ◽  
K. R. Pearson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Torsion Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
List Type ◽  
Type Number ◽  
Quaternion Group ◽  
Group Of Units

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

A Remark on the Units of Finite Order in The Group Ring of a Finite Group

1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Direct Product ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Quaternion Group ◽  
Group Of Units ◽  
A Finite Group

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

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