Quasibases for nonseparable $p$-groups

Abstract

This paper is an extension of the work developed in [4] on quasibases of abelian $p$-groups and based on the doctoral dissertation ofAndrija Vodopivec [5].We introduce the ideas of a $\delta$-combination and height of an inductive quasibasis and show that the height of a quasibasis is invariant for related inductive quasibases. Moreover, an abelian $p$-group is separable if and only if the heights of all $\delta$-combinations are zero. Finally, we show that an abelian $p$-group is not reduced if and only if there exists a $\delta$-combination with infinite height.