On subgroups of fixed index

G. K. White

Vol. 28 (1969), No. 1, 225–232

DOI: 10.2140/pjm.1969.28.225

Abstract

If k ∈𝒦, where 𝒦 is a subgroup of a group 𝒮 , then closure implies k²,k³,⋯,∈𝒦. Nonempty subsets S ⊂𝒮 with the inverse property s^m ∈ S implies s,s²,⋯,s^m ∈ S (m = 1,2,⋯) will be called stellar sets. Let p^α be a fixed prime power. If a stellar set S of an abelian group 𝒮 intersects every subgroup ℋ of index p^α in 𝒮 , and 0∉S, then the cardinal |S| of S is bounded below by p^α (Theorem 3), when 𝒮 satisfies a mild condition.

Mathematical Subject Classification

Primary: 20.30

Milestones

Received: 13 October 1967

Published: 1 January 1969