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

## 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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