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Finite groups with three conjugacy class sizes of some elements

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Abstract

Let G be a finite group. We prove as follows: Let G be a p-solvable group for a fixed prime p. If the conjugacy class sizes of all elements of primary and biprimary orders of G are {1,p a, n} with a and n two positive integers and (p,n) = 1, then G is p-nilpotent or G has abelian Sylow p-subgroups.

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Acknowledgements

The author is very grateful to the referee for reading the manuscript carefully and providing a lot of valuable suggestions and useful comments. It should be mentioned that the final version of this paper could not have been refined so well without his or her outstanding efforts. This research is supported by the National Natural Science Foundation of China (10771132), SGRC (GZ310), the Research Grant of Shanghai University and Shanghai Leading Academic Discipline Project (J50101).

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  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, People’s Republic of China

    QINGJUN KONG

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  1. QINGJUN KONG
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Correspondence to QINGJUN KONG.

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KONG, Q. Finite groups with three conjugacy class sizes of some elements. Proc Math Sci 122, 335–337 (2012). https://doi.org/10.1007/s12044-012-0089-0

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  • DOI: https://doi.org/10.1007/s12044-012-0089-0

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