Vol. 15, No. 2, 1965

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
A generalization of the coset decomposition of a finite group

Basil Gordon

Vol. 15 (1965), No. 2, 503–509
Abstract

Let G be a finite group, and suppose that G is partitioned into disjoint subsets: G = i=1hAi. If the Ai are the left (or right) cosets of a subgroup H G, then the products xy, where x Ai and y Aj, represent all elements of any Ak the same number of times. It turns out that certain other decompositions of G of interest in algebra enioy this same property; we will call such a partition π an α-partition.

In this paper all α-partitions are determined in the case G is a cyclic group of prime order p; they arise by choosing a divisor d of p 1, and letting the Ai be the sets on which the d-th power residue symbol (x∕p)d has a fixed value. It is shown that if an α-partition is invariant under the inner automorphisms of G (strongly normal) then it is also invariant under the antiautomorphism x x1. For such α-partitions (called weakly normal) it is shown that the set Ai containing the identity element is a group. An example shows that this need not hold for nonnormal partitions.

Mathematical Subject Classification
Primary: 20.25
Milestones
Received: 17 April 1964
Published: 1 June 1965
Authors
Basil Gordon