On permutation groups of finite type

A permutation group $G$ is said a group of finite type $k$, $k$ a positive integer, if each non-identity element of $G$ has exactly $k$ fixed points. We show that a group $G$ can be faithfully represented as an irredundant permutation group of finite type if and only if $G$ has a {\nt} {\nr} {\parti} such that each component has finite bounded index in its normalizer. An asymptotic structure theorem for locally (soluble-by-finite) groups of finite type is proved. Finite sharp irredundant permutation groups of finite type, not $p$-groups, are determined.