Non-Abelian sharp permutation \(p\)-groups.
A permutation group of finite degree \(d\) is said to be of type \(\{k\}\) (\(k\) a non-negative integer) if every non-identity element of \(G\) fixes exactly \(k\) points. \(G\) is a sharp permutation group of type \(\{k\}\) if it is of type \(\{k\}\) and \(|G|=d-k\). By the Orbit Counting Lemma, this is equivalent to a group of type \(\{k\}\) with exactly \(k+1\) orbits. For instance, a group with \(k\) global fixed points and one regular orbit is a sharp permutation group of type \(\{k\}\). A sharp permutation group of type \(\{k\}\) is called irredundant if it has no global fixed points and no regular orbits. The author has already described the sharp irredundant permutation groups of finite type, except for the case of non-Abelian \(p\)-groups, in preceding papers. This paper completes this description, by giving properties of finite non-Abelian \(p\)-groups with a faithful representation as a sharp irredundant group of type \(\{k\}\). The author concludes by constructing examples, describing precisely those \(p\)-groups with order at most \(p^5\) (\(p\neq 2\)), and determining the possible values of \(k\) in that case.
sharp permutation groups of type \(\{k\}\)
scientific article; zbMATH DE number 2114526