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 ''p''^''n'' of ''G'' then the minimal polynomial is of the form (''X'' − 1)^''r'' for some ''r'' ≤ ''p''^''n''. The Hall–Higman theorem states that one of the following 3 possibilities holds:
*''r'' = ''p''^''n''
*''p'' is a Fermat prime and the Sylow 2-subgroups of ''G'' are non-abelian and ''r'' ≥ ''p''^''n'' −''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 2^''n'' and ''r'' ≥ 2^''n'' − 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

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{{DEFAULTSORT:Hall-Higman theorem
Theorems in group theory
Number theory