Journal of the Ramanujan Mathematical Society

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𝓏-Classes and Rational Conjugacy Classes in Alternating Groups


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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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  • 𝓏-Classes and Rational Conjugacy Classes in Alternating Groups

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Authors

Sushil Bhunia
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
Dilpreet Kaur
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
Anupam Singh
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India

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 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.

References