scholarly journals A Classification of DCI(CI)-Subsets for Cyclic Group of Odd Prime Power Order

2000 ◽  
Vol 78 (1) ◽  
pp. 24-34 ◽  
Author(s):  
Qiongxiang Huang ◽  
Jixiang Meng
Keyword(s):  
Cyclic Group ◽  
Prime Power ◽  
Prime Power Order

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

The classification of minimal non-prime-power order groups

1992 ◽  
Vol 59 (6) ◽  
pp. 521-524 ◽  
Author(s):  
Joseph A. Gallian ◽  
David Moulton
Keyword(s):  
Prime Power ◽  
Prime Power Order

Finite Groups with Some Subgroups of Given Order

2013 ◽  
Vol 20 (03) ◽  
pp. 457-462 ◽  
Author(s):  
Jiangtao Shi ◽  
Cui Zhang ◽  
Dengfeng Liang
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Solvable Groups ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Prime Power Order

Let [Formula: see text] be the class of groups of non-prime-power order or the class of groups of prime-power order. In this paper we give a complete classification of finite non-solvable groups with a quite small number of conjugacy classes of [Formula: see text]-subgroups or classes of [Formula: see text]-subgroups of the same order.

On ∗-clean group rings over abelian groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750152 ◽  
Author(s):  
Dongchun Han ◽  
Yuan Ren ◽  
Hanbin Zhang
Keyword(s):  
Cyclic Group ◽  
Prime Power ◽  
Associative Ring ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary Condition ◽  
Group Rings ◽  
Prime Power Order

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

ON THE CLASSIFICATION OF GROUPS OF PRIME-POWER ORDER BY COCLASS: THE 3-GROUPS OF COCLASS 2

2013 ◽  
Vol 23 (05) ◽  
pp. 1243-1288 ◽  
Author(s):  
BETTINA EICK ◽  
C. R. LEEDHAM-GREEN ◽  
M. F. NEWMAN ◽  
E. A. O'BRIEN
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Prime Power Order ◽  
Computational Tools ◽  
Significant Step ◽  
New Phenomena ◽  
Full Classification

In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification. On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.

scholarly journals A computer aided classification of certain groups of prime power order: Corrigendum

1977 ◽  
Vol 17 (2) ◽  
pp. 317-319
Author(s):  
Judith A. Ascione ◽  
George Havas ◽  
C.R. Leedham-Green
Keyword(s):  
Theoretical Analysis ◽  
Prime Power ◽  
Accurate Description ◽  
Maximal Class ◽  
Prime Power Order ◽  
Computer Aided

The first four paragraphs of [1, p. 258] are a mildly erroneous over simplification of the situation. A more accurate description follows.The analysis of two-generator 3-groups of second maximal class goes along the following lines. We first define a class of group whose structure is particularly amenable to theoretical analysis.

scholarly journals A computer aided classification of certain groups of prime power order

1977 ◽  
Vol 17 (2) ◽  
pp. 257-274 ◽  
Author(s):  
Judith A. Ascione ◽  
George Havas ◽  
C. R. Leedham-Green
Keyword(s):  
Prime Power ◽  
Maximal Class ◽  
Prime Power Order ◽  
Computer Aided ◽  
Low Order

A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 38 are described, and in some cases with orders up to 310. The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance.

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