×

On groups with infinite almost normal subgroups. (English. Russian original) Zbl 0572.20024

Sov. Math. 27, No. 10, 73-81 (1983); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1983, No. 10(257), 57-63 (1983).
The subgroup H of a group G is almost normal in G if it has only finitely many conjugates in G, or equivalently, if its normaliser has finite index in G. The authors study groups in which all infinite subgroups are almost normal. If all subgroups, finite and infinite, are almost normal, the group is a FIZ group, that is a finite extension of its centre [the reviewer, Math. Z. 63, 76-96 (1955; Zbl 0064.25201)]. Thus it only remains to study groups in which all infinite, but not all finite, subgroups are almost normal. The authors classify these groups under the additional assumption that if the group is not of Černikov type, then it is locally almost decomposable. The groups that turn up are certain extensions of infinite abelian groups by finite groups acting, by conjugation, without invariant subgroups, and extensions of finite groups by such groups.
{A few minor misprints in the Russian original have been faithfully copied in the translation. In the list of references the 1967 Moscow edition of A. G. Kurosh’s ”Theory of Groups” [Zbl 0189.308] is quaintly described as a Russian translation.}
Reviewer: B.H.Neumann

MSC:

20F24 FC-groups and their generalizations
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups
PDFBibTeX XMLCite