Hall–Higman theorem

In

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Statement

Suppose that G is a p-solvable group with no normal p-subgroups,

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
  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

• MR 0569209
• 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,
  MR 0072872