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Groups with normality conditions for non-periodic subgroups. (English) Zbl 1238.20043

This article deals with the influence of the non-periodic subgroups contained in a (non-periodic) group. Sections 2 and 3 are on embedding properties like subnormality: If all non-periodic subgroups are subnormal of defect at most \(k\), then every subgroup is subnormal of defect \(k\) and the group is nilpotent of class bounded by a function of \(k\) (Theorem 2.1). If there is no bound on the defect, the group is a Baer group (Lemma 2.2). If every finitely generated non-nilpotent subgroup of \(G\) has a finite non-nilpotent homomorphic image and all non-periodic subgroups are pronormal, then \(G\) is Abelian (Theorem 2.6). If all non-periodic subgroups are normal-by-finite, then the same is true for every subgroup (Theorem 3.3). – Section 4 deals with locally graded groups and group classes \(\mathcal C\) such that \(G\) belongs to \(\mathcal C\) whenever all proper non-periodic subgroups belong to \(\mathcal C\). Classes of this kind are supersoluble, polycyclic, soluble \(\pi\)-minimax, locally soluble groups and a family of classes given in Corollary 4.2.

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20F22 Other classes of groups defined by subgroup chains
20F19 Generalizations of solvable and nilpotent groups
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