On subgroups of prime power index

Langdon Frank Harris

Vol. 35 (1970), No. 1, 117–126

DOI: 10.2140/pjm.1970.35.117

Abstract

Let G be an abelian group. A set S ⊂ G is a stellar set if mx ∈ S implies x,2x,⋯,mx ∈ S. Let p^α be a fixed prime power. It is shown that if S ∩ p^αG = ∅,G satisfies a mild condition, and S intersects all the subgroups K of index G : K = p^α, then the cardinality of S is bounded below by p^α + p^α−1. This bound is the best possible. The problem is reduced to solving a number of congruence relations

with lattice points (x₁,x₂,⋯,x_n) in a stellar set S in Euclidean n-space. This in turn leads to an interesting result on congruence classes of subgroups and points which tells something about the solution in integers of the above congruence relation.

Mathematical Subject Classification

Primary: 20.30

Milestones

Received: 16 June 1969

Published: 1 October 1970