Transitive generalized toggle groups containing a cycle

Abstract

In \cite{striker2018rowmotion} Striker generalized Cameron and Fon-Der-Flaass's notion of a toggle group. In this paper we begin the study of transitive generalized toggle groups that contain a cycle. We first show that if such a group has degree $n$ and contains a transposition or a 3-cycle then the group contains $A_n$. Using the result about transpositions, we then prove that a transitive generalized toggle group that contains a short cycle must be primitive. Employing a result of Jones \cite{jones2014primitive}, which relies on the classification of the finite simple groups, we conclude that any transitive generalized toggle group of degree $n$ that contains a cycle with at least 3 fixed points must also contain $A_n$. Finally, we look at imprimitive generalized toggle groups containing a long cycle and show that they decompose into a direct product of primitive generalized toggle groups each containing a long cycle.