Vol. 15, No. 2, 1965

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A generalization of the coset decomposition of a finite group

Basil Gordon

Vol. 15 (1965), No. 2, 503–509

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
Received: 17 April 1964
Published: 1 June 1965
Basil Gordon