Abelian sharp permutation groups. (Q1770484)

A permutation group \(G\) of finite degree \(n\) is said to be of finite type \(\{k\}\) if every \(1\neq g\in G\) fixes exactly \(k\) letters for some nonnegative integer \(k\). \(G\) is called a sharp permutation group of type \(\{k\}\) if \(G\) is of finite type \(\{k\}\) and \(|G|=n-k\); \(G\) is called an irredundant (permutation) group of type \(\{k\}\) if it is a group of type \(\{k\}\) without global fixed-points and regular orbits. In the present paper a method is given to construct all faithful representations of finite Abelian groups \(G\) as sharp irredundant permutation groups of type \(\{k\}\), where \(k\) is a positive integer. Such groups \(G\) are elementary Abelian \(p\)-groups by a result of the author [Eur. J. Comb. 22, No. 6, 821-837 (2001; Zbl 0989.20001)]. Certain partitions of \(G\), called \(\alpha\)-coverings, are relevant for the construction given in Theorem 1 of the paper.