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On the influence of transitively normal subgroups on the structure of some infinite groups. (English) Zbl 1293.20028
A group is called T-group, if normality and subnormality coincide, it is generalized radical, if it has a normal series with locally nilpotent or locally finite quotients. A subgroup of a group is a transitively normal subgroup if it is normal in every subgroup in which it is subnormal. This article describes the neighbourhood of the first and the last named concept if many subgroups are concerned.
Theorem A: If $$G$$ is a locally finite group whose cyclic subgroups of prime order and of order $$4$$ are transitively normal, then $$G$$ is hypercyclic and the locally nilpotent residual $$L$$ of $$G$$ is an Abelian Hall subgroup such that all subgroups of $$L$$ are invariant in $$G$$.
If $$G$$ is locally finite and all primary cyclic subgroups are transitively normal, then all subgroups of $$G$$ are T-groups (Corollary A 1).
If, instead, $$G$$ is non-periodic, non-Abelian and generalized radical and all cyclic subgroups are transitively normal, then $$G$$ possesses an Abelian subgroup $$L$$ such that $$|G:L|=2$$ and an element $$b\in G\setminus L$$ operates on $$L$$ by conjugation as inversion (Theorem B).

##### MSC:
 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth 20F19 Generalizations of solvable and nilpotent groups 20F50 Periodic groups; locally finite groups 20F14 Derived series, central series, and generalizations for groups
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