Hall–Higman theorem

In mathematical group theory, the Hall-Higman theorem, due to, 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 ^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 is 3-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 ² with r=3−1.