Character tables of p-groups with derived subgroup of prime order III

Abstract

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. In this paper we compare the number of non-isomorphic p-groups with derived subgroup of order p with the number of non-isomorphic character tables of these groups. We show that the difference between the number of non-isomorphic groups of this type and the number of non-isomorphic character tables of these groups increases exponentially. Furthermore, we prove that if we fix the index of the center of these groups, say p 2 m , and we let the size of the groups grow bigger, then, for each character table there are on average ( 2 m ) ! / ( 2 m m ! ) non-isomorphic groups whose character table are isomorphic to the given one.