Hall–Higman theorem
In mathematical group theory, the Hall–Higman theorem, due to {{harvs|txt|authorlink=Philip Hall|first=Philip|last=Hall|first2=Graham|last2=Higman|author2-link=Graham Higman||year=1956|loc=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 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 −pn−1
- p = 2 and the Sylow q-subgroups of G are non-abelian for some Mersenne prime q = 2m − 1 less than 2n and r ≥ 2n − 2n−m.
Examples
References
- {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | url=https://www.ams.org/bookstore-getitem/item=CHEL-301-H | publisher=Chelsea Publishing Co. | location=New York | edition=2nd | isbn=978-0-8284-0301-6 | mr=569209 | year=1980}}
- {{citation|last=Hall|first= P.|last2= Higman|first2= Graham|year=1956|title=On the p-length of p-soluble groups and reduction theorems for Burnside's problem|doi=10.1112/plms/s3-6.1.1 |mr=0072872|journal=Proceedings of the London Mathematical Society |series=Third Series|pages=1–42|volume=6}}
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