Concerning uniqueness-bases of finite groups with applications to 𝑝-groups of class 2

Abstract

A uniqueness-basis (U-basis) of a finite group G has been defined as ordered set of elements Qi, Q2, . , Q, such that every element of G can be expressed uniquely in the QllxQ2X2 . . . Q,xp, where each xi is a least positive residue modulo the order of Qi.t In the case of the abelian groups the notion of U-basis is of undisputed importance: the theorem that every finite abelian group A has a U-basis may fairly be regarded as the cornerstone of the theory of abelian groups. In the case of most non-abelian groups, however, the concept of U-basis is of doubtful advantage, especially in the general above. Of greater usefulness, naturally, would be a simplest type of U-basis for the group under consideration. But the problem of constructing a definition of a form which shall be significant for reasonably general categories-the non-abelian p-groups, say-is an exceedingly difficult one. We offer a tentative definition in the case of the regular p-groups? (??2-3), which have, in common with the abelian p-groups, the property that the orders of the elements in every U-basis constitute a set of invariants of the group. In ?4 we shall show how a normal U-basis may be used in constructing for every regular p-group G of class 2 a simply-isomorphic representation by l-matrices? -matrices whose coordinates are residue classes modulo certain powers of p. These representations of G are of interest in that they usually involve a much smaller number of rows than do the matrix-representations whose coordinates are in a field. (The i-matrices are by no means novel; they have long been used for representing automorphisms of abelian p-groups.) In ?5 we shall discuss the representation by i-matrices of the group of isomorphisms of G, and in ??6-7 we shall describe, very briefly, a representation of G as a multiplicative group in a finite ring. 1. In this section we shall state-for the most part without proof-several theorems which afford a set of criteria for the existence of a U-basis in a finite group G.