Subdirect sums and infinite Abelian groups

k-Sums in Abelian Groups

Given a finite subset A of an abelian group G, we study the set k ∧ A of all sums of k distinct elements of A. In this paper, we prove that |k ∧A| > |A| for all k ∈ {2, . . . , |A| − 2}, unless k ∈ {2, |A| − 2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite sets A ⊆ G for which |k ∧ A| = |A| for some k ∈ {2, . . . , |A| − 2}. This result answers a q...

Subdirect Sums of Rings

Finite Abelian Groups and Character Sums

These are some notes I have written up for myself. I have compiled what I feel to be some of the most important facts concerning characters on finite abelian groups, especially from a number theoretic point of view. I decided to distribute these because I couldn’t find a single reference that covered everything that I thought was relevant to the material I will discuss on Wednesday. Of course, ...

Sums and k-sums in abelian groups of order k

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a1, . . . , ar that does not have 0 as a k-sum is attained at the sequence b1, . . . , br−k+1,0, . . . ,0, where b1, . . . , br−k+1 is chose...

Counting subset sums of finite abelian groups

In this paper, we obtain an explicit formula for the number of zero-sum k-element subsets in any finite abelian group.