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 → x−1. 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.
