> < ^ Date: Wed, 11 Oct 2000 12:17:57 +0100
> < ^ From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de >
> < ^ Subject: Re: Frobenius

Dear Gap Forum,

Kurt Ewald asked:

Does anybody know a program that tests, whether a finite Group is a
frobenius group?

To the best of my knowledge there is no function in the GAP
distribution that would directly answer the question whether a given
finite group is a Frobenius group. However if you look up a
group-theory text book, e.g. Huppert, Endliche Gruppen I, Chapter V,
Paragraph 8, p. 495 ff 'Frobeniusgruppen' you will find many necessary
conditions for a group to be a Frobenius group that can easily be
tested using GAP, e.g. (8.7) that the Sylow subgroups must be cyclic
or generalized quaternion groups. Which of these you should use you
may have to decide according to the way your group is given. If all
these are fulfilled the (by 8.17 unique) Frobenius kernel can fairly
easily be detected and the (by definition sufficient) condition that
the complements of the Frobenius kernel have trivial intersection be

Hope this helps, kind regards Joachim Neubueser

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