Normal subgroups and projectivities of finite groups
Let G be a finite group, \(\sigma\) a projectivity of G onto \(G^{\sigma}\), H a normal subgroup of G and \(T^{\sigma}\) the core of \(H^{\sigma}\) in \(G^{\sigma}\). The author investigates the structure of H/T. He proves the following theorem: Let G be a finite p-group (p an odd prime), G/H cyclic, \(\sigma\) index-preserving and T trivial; then H is abelian. He remarks that no counterexamples to the case \(p=2\) are known. As a corollary of the main theorem, the author proves that if G is a group of odd order, then H/T is abelian. Examples of groups which satisfy the hypotheses of the theorem with \(H^{\sigma}\) nonabelian are given in the last section, using previous examples of groups with core- free quasinormal subgroups [\textit{J. G. Thompson}, Math. Z. 96, 226-227 (1967; Zbl 0189.317)].
scientific article; zbMATH DE number 3929262
scientific article