Straight Bases of Abelian p-Groups

Abstract

Gauss, already in his “Disquisitiones arithmeticae” (1801) seems to have recognized, though implicitly, the importance of the basis concept in finite abelian groups. In the theory of infinite abelian groups, L. Fuchs, making use of basic subgroups, developed the useful notion of a quasibasis of a p-group. In this article, we propose another concept involving generating sets with certain uniqueness properties which we call straight bases. Among the properties of a straight basis, we single out the following:

• 1. If B is a straight basis of a p-group A, then any non-zero element of A can be expressed uniquely as a linear combination of elements of B with non-negative integer coefficients smaller than p.

• 2. The relations between the elements of B give rise to a family of integers called an s-factor set which determines the group A up to isomorphism.

The first author holds the Canadian C.R.S.N.G. grant No. A5591.

The most part of this research was done while the second author was an invited visitor at the University of Montréal in the summer 1982.