A characterization of some metacyclic groups
Eugene Schenkman · Proceedings of the American Mathematical Society · 1957
Szasz [1 ] has recently shown that a group is cyclic if and only if it satisfies condition (A) below. (A) Every cyclic subgroup of the group is for some positive integer k the subgroup generated by the kth powers of the elements of the group. We shall extend this idea here to show that a metacyclic group whose commutator subgroup has order relatively prime to its index is characterized as a solvable group satisfying condition (B) below. (B) Every member of a composition series (i.e. every subinvariant subgroup) is for some positive integer k the subgroup generated by the kth powers of the elements of the group. (If G denotes the group, the subgroup will be denoted by G(k)). If the hypothesis of solvability is not included it is easy to check that many completely reducible groups and their extensions, (including, for example, all simple groups and all symmetric groups) satisfy condition (B). We were unable to characterize these. We first list some of the properties of a group satisfying condition (B). (1) Every subinvariant subgroup is a fully invariant subgroup of G; that is, it is mapped into itself by all endomorphisms of G. For this is true for all G(k). (2) Every homomorphic image of a group satisfying (B) itself also satisfies (B). For the generators of corresponding normal subgroups are kth