×

The conjugacy classes of a subgroup \(S^m_n:C_m\) of \(S_{mn}\), prime \(m\). (English) Zbl 1146.20300

Summary: The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let \(n,m\) be positive integers and \(S^m_n\) be the direct product of \(m\) copies of the symmetric group \(S_n\) of degree \(n\). Then \(S^m_n\) is a subgroup of the symmetric group \(S_{mn}\) of degree \(m\times n\). Let \(g\in S^m_n\), of type \([m^n]\) where each \(m\)-cycle contains one symbol from each set of symbols in that order on which the copies of \(S_n\) act. Then \(g\) permutes the elements of the copies of \(S_n\) in \(S^m_n\) and generates a cyclic group \(C_m=\langle g\rangle\) of order \(m\). The wreath product of \(S_n\) with \(C_m\) is a split extension or semi-direct product of \(S^m_n\) by \(C_m\), denoted by \(S^m_n:C_m\). It is clear that \(S^m_n:C_m\) is a subgroup of the symmetric group \(S_{mn}\).
In this paper we give a method similar to coset analysis for constructing the conjugacy classes of \(S^m_n:C_m\), where \(m\) is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group \(S^m_n:C_m\), this method is useful in the construction of Fischer-Clifford matrices of the group \(S^m_n:C_m\). These Fischer-Clifford matrices are useful in the construction of the character table of \(S^m_n:C_m\).

MSC:

20B35 Subgroups of symmetric groups
20E45 Conjugacy classes for groups
20C30 Representations of finite symmetric groups
20C40 Computational methods (representations of groups) (MSC2010)
20E22 Extensions, wreath products, and other compositions of groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Ali M. I. M., Math. Japonica 38 pp 155–
[2] DOI: 10.1090/S0025-5718-1994-1208219-2 · doi:10.1090/S0025-5718-1994-1208219-2
[3] DOI: 10.2307/1968599 · Zbl 0017.29705 · doi:10.2307/1968599
[4] Fischer B., Progr. Math. 95 pp 1– (1991)
[5] Isaac I. M., Character Theory of Finite Groups (1976)
[6] Rotman J. J., A First Course in Abstract Algebra (1996) · Zbl 0847.00004
[7] Kerber A., Lecture Notes in Mathematics 240, in: Representations of Permutation Groups I, Vol. 1 (1971) · Zbl 0232.20014 · doi:10.1007/BFb0067943
[8] DOI: 10.1090/S0025-5718-99-01157-6 · Zbl 0962.20001 · doi:10.1090/S0025-5718-99-01157-6
[9] DOI: 10.1080/16073606.1999.9632080 · Zbl 0944.20010 · doi:10.1080/16073606.1999.9632080
[10] DOI: 10.2989/16073600509486123 · Zbl 1087.20012 · doi:10.2989/16073600509486123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.