The Exceptional Hall Numbers

Abstract

A positive integer m is called a Hall number if any finite group of order precisely divisible by m has a Hall subgroup of order m. We prove that, except for the obvious examples, the three integers 12, 24 and 60 are the only Hall numbers, solving a problem proposed by Jiping Zhang.

Keywords

Hall subgroups / Hall numbers / Finite groups

References

[1] BrayJN, HoltDF, Roney-DougalCMThe Maximal Subgroups of the Low-Dimensional Finite Classical Groups, 2013New YorkCambridge University Press.

[2] Conway, J.H., Curtis, R.T., Noton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

[3] GorensteinD. The characterization of finite groups with dihedral Sylow 2-subgroups, I.. J. Algebra, 1965, 2185-151.

[4] GorensteinDFinite Simple Groups: An Introduction to Their Classification, 1982New YorkPlenum Press.

[5] RobinsonDJA Course in the Theory of Groups, 1982BerlinSpringer.

[6] SerreJ-PA Course in Arithmetic, 1973BerlinSpringer.

Funding

National Natural Science Foundation of China(11931005)

RIGHTS & PERMISSIONS

Peking University