Journal of the Ramanujan Mathematical Society

Volume 34, Issue 2, June 2019 pp. 169–183.

z-classes and rational conjugacy classes in alternating groups

Authors: Sushil Bhunia, Dilpreet Kaur and Anupam Singh
Author institution:IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411 008, India

Summary: 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 An 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.

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