×

Abnormal subgroups and Carter subgroups in some infinite groups. (English) Zbl 1092.20027

Summary: Some properties of abnormal subgroups in generalized soluble groups are considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it is proven that a subgroup \(H\) of a radical group \(G\) is abnormal in \(G\) if and only if every intermediate subgroup for \(H\) coincides with its normalizer in \(G\). This result extends to radical groups the well-known criterion of abnormality for finite soluble groups due to D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugation are also obtained.

MSC:

20E34 General structure theorems for groups
20F19 Generalizations of solvable and nilpotent groups
20F22 Other classes of groups defined by subgroup chains
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
PDFBibTeX XMLCite
Full Text: arXiv