Hall–Higman theorem explained

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 (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 -p^n-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.