Abnormal subgroups and Carter subgroups in some infinite groups

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 on 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 have been also obtained.