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.
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