Comments on Y. O. Hamidoune's Paper "Adding Distinct Congruence Classes"

TL;DR

This paper critiques and corrects a previous proof regarding the number of subset sums in cyclic groups, extending the main result to all finite abelian groups.

Contribution

It identifies a flaw in the original proof, provides a corrected argument, and generalizes the main theorem to all finite abelian groups.

Findings

The original lemma is false for even order cyclic groups.

The main result holds for all finite abelian groups.

A corrected proof confirms the theorem's validity beyond cyclic groups.

Abstract

The main result in Y.~O.~Hamidoune's paper "Adding Distinct Congruence Classes" ({\em Combin.~Probab.~Comput.}~{\bf 7} (1998) 81-87) is as follows: If $S$ is a generating subset of a cyclic group $G$ such that $0 \not \in S$ and $|S| \geq 5$, then the number of sums of the subsets of $S$ is at least $\min (|G|, 2|S| )$. Unfortunately, argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.