Totally inert subgroups of a rank two group constructed by Zassenhaus
Abstract
A subgroup H of an abelian group G is totally inert if, for every non-zero endomorphism φ of G, H is commensurable with φ(H), that is, H ∩ φ(H) has finite index in H and in φ(H). In this paper we provide necessary and sufficient conditions for the existence of rank two subgroups which fail to be totally inert of a particular torsion-free group of rank two G such that End(G) = Z[i], the ring of Gaussian integers, obtained by a classical construction of Zassenhaus. The results obtained here partially solve a problem raised in a recent paper by Brendan Goldsmith and the author, where totally inert subgroups of general abelian groups are investigated.
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