# Frobenius group

A permutation group $G$ on a set $X$ is Frobenius if no non-trivial element of $G$ fixes more than one element of $X$. Generally, one also makes the restriction that at least one non-trivial element fix a point. In this case the Frobenius group is called non-regular.

The stabilizer of any point in $X$ is called a Frobenius complement, and has the remarkable property that it is distinct from any conjugate by an element not in the subgroup. Conversely, if any finite group $G$ has such a subgroup, then the action on cosets of that subgroup makes $G$ into a Frobenius group.

Title Frobenius group FrobeniusGroup 2013-03-22 13:16:30 2013-03-22 13:16:30 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 20B99 Frobenius complement