Fixed point free automorphisms of groups related to finite fields

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Fixed point free automorphisms of groups related to finite fields

Let G = Fqo 〈β〉 be the semidirect product of the additive group of the field of q = p elements and the cyclic group of order d generated by the invertible linear transformation β defined by multiplication by a power of a primitive root of Fq. We study endomorphisms of G. We find an arithmetic condition on d so that every endomorphism is determined by its values on (1, 1) and (0, β). When that i...

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Finite Groups with Fixed-point-free Automorphisms of Prime Order.

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Automorphisms of Free Groups Have Finitely Generated Fixed Point Sets

An automorphism of a finitely generated free group F extends to a homeomorphism of the end completion E of F. The set of fixed points of this homeomorphism is finitely generated in a certain sense. In particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense. The emphasis on E instead of F in this paper is analogous to Thurston’s st...

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Finite Fixed Point Free Automorphism Groups

Preface A famous theorem by Frobenius in 1901 proves that if a group G contains a proper non trivial subgroup H such that H ∩ g −1 Hg = {1 G } for all g ∈ G \ H, then there exists a normal subgroup N such that G is the semidirect product of N and H. Groups with this property-the so called Frobenius groups-arise in a natural way as transitive permutation groups, but they can also be characterize...

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ژورنال

عنوان ژورنال: Finite Fields and Their Applications

سال: 2012

ISSN: 1071-5797

DOI: 10.1016/j.ffa.2012.03.002