Equivalent Condition of Semiregular

todo

Finite Permutation Groups with a Transitive Minimal normal subgroup, Lemma 4.2

Let $\Omega$ be a finite set. Then

• A subgroup of $\mathrm{Sym}(\Omega )$ is semiregular iff its full centralizer is transitive.
• The full centralizer of a transitive subgroup of $\mathrm{Sym}(\Omega )$ is semiregular.

\begin{proof} Suppose that $C:=C_{\mathrm{Sym}(\Omega )}(G)$ is transitive. Then for any $\alpha ,\beta \in \Omega$, there is $g\in C$ such that ${\alpha }^g=\beta$. Then we have $G_{\alpha }=G_{\alpha }^g=G_{{\alpha }^g}=G_{\beta }$ for any $\alpha ,\beta \in \Omega$. Since $G$ is faithful on $\Omega$, we have $G_{\alpha }=\{1\}$ for all $\alpha \in \Omega$.

Now suppose that $G$ is semiregular.

\end{proof}