Hall–Higman theorem

In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman (1956, Theorem B), describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group.

Statement

Suppose that G is a p-solvable group with no normal p-subgroups, acting faithfully on a vector space over a field of characteristic p. If x is an element of order pⁿ of G then the minimal polynomial is of the form (X − 1)^r for some r ≤ pⁿ. The Hall–Higman theorem states that one of the following 3 possibilities holds:

• r = pⁿ
• p is a Fermat prime and the Sylow 2-subgroups of G are non-abelian and r ≥ pⁿ −pⁿ⁻¹
• p = 2 and the Sylow q-subgroups of G are non-abelian for some Mersenne prime q = 2^m − 1 less than 2ⁿ and r ≥ 2ⁿ − 2^n−m.

Examples

The group SL₂(F₃) is 3-solvable (in fact solvable) and has an obvious 2-dimensional representation over a field of characteristic p=3, in which the elements of order 3 have minimal polynomial (X−1)² with r=3−1.

References

• Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 569209 
• Hall, P.; Higman, Graham (1956), "On the p-length of p-soluble groups and reduction theorems for Burnside's problem", Proceedings of the London Mathematical Society. Third Series 6: 1–42, doi:10.1112/plms/s3-6.1.1, MR 0072872