On involutions arising from graphs

Abstract

We investigate various aspects of involutions of groups, i.e. antiautomorphisms of order at most two. The emphasis is on finite Abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated Abelian groups with identity as the involution. The present paper is motivated by our research into switching classes of combinatorial graphs where the edges have skew gains.