# One-prime power hypothesis for conjugacy class sizes

Document Type : Research Paper

Authors

1 University of East Anglia

2 Fitzwilliam College, University of Cambridge

Abstract

A finite group $G$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to $S \times A$ where $A$ is abelian and $S \cong PSL_2(q)$ for $q \in {4,8}$. We confirm this conjecture.

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#### References

[1] R. Abbott et al., Atlas of Finite Group Representations - Version 3, brauer.maths.qmul.ac.uk/Atlas/v3/.
[2] A. R. Camina, Finite groups of conjugate rank 2, Nagoya Math. J., 53 (1974) 47–57.
[3] A. R. Camina and R. D. Camina, Implications of conjugacy class size, J. Group Theory, 1 (1998) 257–269.
[4] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a survey, Asian-Eur. J. Math., 4 (2011) 559–588.
[5] A. L. Delgado and Y.-F. Wu, On locally finite groups in which every element has prime power order, Illinois J. Math., 46 (2002) 885-891.
[6] D. Gorenstein, Finite Groups, Harper Row, Publishers, New York-London, 1968.
[7] M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95–105.
[8] I. M. Isaacs, Groups with many equal classes, Duke Math. J., 37 (1970) 501–506.
[9] N. Itˆ o, On finite groups with given conjugate types. I, Nagoya Math. J., 6 (1953) 17–28.
[10] L. S. Kazarin, On groups with isolated conjugacy classes, Izv. Vyssh. Uchebn. Zaved. Mat., 7 (1981) 40–45.
[11] M. L. Lewis, Solvable groups having almost relatively prime distinct irreducible character degrees, J. Algebra, 174 (1995) 197–216.
[12] G. H. Qian, Finite groups with many elements of prime order, J. Math. (Wuhan), 25 (2005) 115–118.
[13] J. Rebmann, F-Gruppen, Arch. Math. (Basel), 22 (1971) 225–230.
[14] B. Taeri, Cycles and bipartite graph on conjugacy class of groups, Rend. Semin. Mat. Univ. Padova, 123 (2010) 233–247.