Permutation representations with trivial set-stabilizers
The author calls a subgroup C of a finite group G abnormal if there is a subset X of G such that \(XC=X\) and yX\(\neq X\) for all \(y\neq 1\) in G. Such a subgroup cannot contain a nontrivial normal subgroup of G, and it is easily seen that C is abnormal if and only if it is a point-stabilizer of a transitive permutation representation of G on a set \(\Omega\), say, and the set-stabilizer of some subset \(\Gamma\) of \(\Omega\) is 1. The main result of the paper shows that cyclic subgroups are usually abnormal. Theorem: If C is a cyclic subgroup of G, and C contains no nontrivial normal subgroup of G, then either C is abnormal or G is a nondihedral Frobenius group of order 42 or 20, a dihedral group of order 10 or 8, a symmetric group of degree 3 or 4, or an alternating group of degree 4. This theorem has an application to the problem of realizing algebraic number fields; see Theorem 2.13 of \textit{R. S. Pierce} and \textit{C. I. Vinsonhaler} [in ''Honolulu Symp. Abelian Groups'', Lect. Notes Math. 1006, 49-96 (1983; Zbl 0515.12006)]. The author points out that a different proof of his main result in the case that G has odd order appears in [\textit{D. Gluck}, Can. J. Math. 35, 59-67 (1983; Zbl 0509.20002)].
scientific article; zbMATH DE number 3957399