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 Sn, 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 Sn 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 1 (and 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 An is same as the number of conjugacy classes which are rational.