Transitive generalized toggle groups containing a cycle

In (Striker in Discret Math Theor Comput Sci 20, 2018), 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 (Bull Aust Math Soc 89(1):159-165, 2014), 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.