Fixed point ratios in actions of finite classical groups, II

Isolated Blocks in Finite Classical Groups, II

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...

Grassmannian Fixed Point Ratios

We provide estimates for the xed point ratios in the permutation representations of a nite classical group over a eld of order q on k-subspaces of its natural n-dimensional module. For suuciently large n each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q ?dk where d is the codime...

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

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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...