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On subgroups of prime power index Harris, L. F.

Abstract

Let G he the direct sum of n ≥ 2 copies of the cyclic group, Z , of integers. Let p be a fixed prime and α ≥ 1 a fixed integer. Consider the subgroups, λ , of G of index p[superscript]α in G . Let S be a subset of G. We say S is a stellar set if ax ε S implies (1.1) x,2x,...,ax ε S for any x ε G and any integer a ≥ 1 . Suppose S is a stellar set, p[superscript]α G ∩ S = ∅, and S intersects all the subgroups λ of G of index p[superscript]α in G . We shall show that then |S| ≥ p[superscript]α + P[superscript]α-1.

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