Affine transformations of finite vector spaces with large orders or few cycles

TL;DR

This paper classifies affine transformations of finite vector spaces with large order or few cycles, and applies these results to classify certain finite primitive permutation groups of affine type.

Contribution

It provides a classification of affine transformations with large order or few cycles and applies this to classify related primitive permutation groups.

Findings

Classified affine transformations of order ≥ p^d/4.

Determined primitive permutation groups of affine type with permutations of order ≥ n/4.

Classified primitive groups of affine type with permutations having at most four cycles.

Abstract

Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least p^d/4, and apply this classification to determine the finite primitive permutation groups of affine type, and of degree n, that contain a permutation of order at least n/4. Using this result we obtain a classification of finite primitive permutation groups of affine type containing a permutation with at most four cycles.