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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

checked.

Hope this helps, kind regards Joachim Neubueser

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