Hall–Higman theorem

In mathematical group theory, the HallHigman 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 HallHigman theorem states that one of the following 3 possibilities holds:

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 (X1)2 with r=31.

References

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