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:
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.