Vol. 35, No. 1, 1970

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On subgroups of prime power index

Langdon Frank Harris

Vol. 35 (1970), No. 1, 117–126

Let G be an abelian group. A set S G is a stellar set if mx S implies x,2x,,mx S. Let pα be a fixed prime power. It is shown that if S pαG = ,G satisfies a mild condition, and S intersects all the subgroups K of index G : K = pα, then the cardinality of S is bounded below by pα + pα1. This bound is the best possible. The problem is reduced to solving a number of congruence relations

λ1x1 + λ2x2 + ⋅⋅⋅+ λnxn ≡ 0(pα)

with lattice points (x1,x2,,xn) 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
Received: 16 June 1969
Published: 1 October 1970
Langdon Frank Harris