The one-prime power hypothesis for conjugacy classes restricted to normal subgroups and quotient groups

Document Type: Research Paper

Author

Fachgruppe Mathematik und Informatik, BU Wuppertal, Wuppertal, Germany

Abstract

We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers‎. ‎Insoluble groups which satisfy the one-prime power hypothesis have been classified‎. ‎However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups‎. ‎In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups‎.

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[1] A. R. Camina and R. D. Camina, One-prime p ower hyp othesis for conjugacy class sizes, Int. J. Group Theory , 6
no. 3 (2017) 13{19.
[2] N. Du and M. L. Lewis, The prime-p ower hyp othesis and solvable groups, Arch. Math. (Basel) , 109 no. 4 (2017)
301-303
[3] Y. Liu and X. Song and J. Zhang, Nonsolvable groups satisfying the prime-p ower hyp othesis, J. Algebra , 442 (2015)
455{483.
[4] B. Taeri, Cycles and bipartite graph on conjugacy class of groups, Rend. Semin. Mat. Univ. Padova , 123 (2010)
233{247.