On Permutation Groups of Finite Type

A permutation group G is said to be 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 non-trivial normal partition 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, notp -groups, are determined.