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A note on the product of conjugacy classes of a finite group

  • Neda AhanjidehEmail author
Article

Abstract

In Guralnick and Moreto (Conjugacy classes, characters and products of elements, arXiv:1807.03550v1, Theorem 4.2) it has been shown that if \(p\ne q\) are two odd primes, \(\pi =\{2,p,q\}\) and G is a finite group such that for every \(\pi \)-elements \(x,y \in G\) with \((O(x),O(y))=1\), \((xy)^G=x^Gy^G\), then G does not have any composition factors of order divisible by pq. In this note, inspired by the above result, we show that if p and q are two primes (not necessarily odd) and G is a finite group such that for every p-element x and q-element \(y \in G\), \((xy)^G=x^Gy^G\), then G does not have any composition factors of order divisible by pq. In particular, we show that if p is an odd prime and G is a finite group such that for every p-element x and 2-element \(y \in G\), \((xy)^G=x^Gy^G\), then G is p-solvable.

Keywords

The product of conjugacy classes Almost simple groups Irreducible character degree 

Mathematics Subject Classification

20E45 20D05 20C15 

Notes

Acknowledgements

The authors would like to thank Professor Alexander Moreto and Professor Robert Guralnick. Thanks also are due to Shahrekord University for financial support (Grant 96GRD1M1023).

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesShahrekord UniversityShahrekordIran

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