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Groups with few normalizer subgroups. (English) Zbl 1129.20025

This paper gives a survey of recent results of the following type. Let \(\mathcal P\) be a group theoretical class or property of subgroups of a group and suppose that \(G\) is a group which has only finitely many \(\mathcal P\)-subgroups \(X_1,X_2,\dots,X_n\) such that \(|G:N_G(X_i)|\) is infinite. What can be said concerning the structure of \(G\)? The same question can be posed if we assume that \(G\) has only finitely many non-\(\mathcal P\)-subgroups whose normalizers have infinite index (of course the possibility that all normalizers of \(\mathcal P\)-subgroups (or non-\(\mathcal P\)-subgroups) have finite index is included). The particular cases in question here are the case when \(\mathcal P\) is the class of Abelian groups and the case when \(\mathcal P\) is the property of being a subnormal subgroup. This very nice paper has a plethora of results on this subject.

MSC:

20F24 FC-groups and their generalizations
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20F22 Other classes of groups defined by subgroup chains
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