A quasi-decomposable abelian group without proper isomorphic quotient groups and proper isomorphic subgroups

John McCormick Irwin and Takashi Ito

Vol. 29 (1969), No. 1, 151–160

DOI: 10.2140/pjm.1969.29.151

Abstract

All of the group in this paper are abelian p-groups without elements of infinite height. A group is said to be quasiindecomposable if whenever H is a summand of G then either H or G∕H is finite. The p-socle of G is the sub-group consisting of all the elements x in G such that px = 0.

In this paper it is shown that there are conditions that can be imposed on the socle of G which are sufficient for G to (a) have no proper isomorphic subgroups; (b) have no proper isomorphic quotient groups; and (c) be quasiindecomposable. Furthermore, it is shown that groups which make these results meaningful actually exist.

Mathematical Subject Classification

Primary: 20.30

Milestones

Received: 22 August 1967

Published: 1 April 1969