Primitive permutation groups containing a cycle

TL;DR

This paper classifies primitive finite permutation groups that contain a cycle, showing that only the alternating and symmetric groups have a cycle fixing at least three points, and discusses historical contributions to this classification.

Contribution

It provides a complete classification of primitive permutation groups containing a cycle, highlighting the unique role of alternating and symmetric groups in this context.

Findings

Only alternating and symmetric groups contain a cycle fixing at least three points.

Classification of primitive permutation groups with a cycle is achieved.

Historical contributions by Jordan and Marggraff are summarized.

Abstract

The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. The contributions of Jordan and Marggraff to this topic are briefly discussed.