Hall polynomials and the fixed space of an automorphism of a finite abelian p-group

Abstract

Recent work on Hall polynomials is used to study the fixed space of a random automorphism of a finite abelian p-group. An expression is found for the chance that an automorphism of an abelian p-group of type \(\lambda \) fixes only the identity. A formula is obtained for the chance that a given subgroup H of type \(\nu \) is the fixed space of an automorphism of an abelian p-group of type (kl). There results generalize work of Rudvalis and Shinoda on the fixed space of an element of GL (n, q).