Another proof of Gluck's theorem (Q1348686)

Let \(G\) be a permutation group acting on a finite set \(\Omega\). A subset \(S\) of \(\Omega\) is called \(G\)-regular if the stabilizer \(\{g\in G\mid Sg=S\}\) is the identity subgroup. \textit{D. Gluck}'s theorem [Can. J. Math. 35, 59-67 (1983; Zbl 0509.20002)] states that \(G\) has a regular subset in \(\Omega\) if \(G\) is of odd order. In the present paper an alternative proof of Gluck's theorem is given which avoids \textit{T. R. Wolf}'s result on primitive solvable groups [Can. J. Math. 34, 1097-1111 (1982; Zbl 0493.20027)] but still relies on the odd order theorem.