z-classes and rational conjugacy classes in alternating groups

Abstract

In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S-n, when n >= 3 and alternating group A(n) when n >= 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S-n is determined by those restricted partitions of n - 2 in which 1 and 2 do not appear as its part. In the case of alternating groups, it is determined by those restricted partitions of n - 3 which has all its parts distinct, odd and in which (1and 2) does not appear as its part, along with an error term. The error term is given by those partitions of n which have distinct parts that are odd and perfect squares. Further, we prove that the number of rational-valued irreducible complex characters for A(n) is same as the number of conjugacy classes which are rational.