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On the structure of the normal subgroups of a group: Nilpotency. (English) Zbl 0759.20011

Let \(G\) be a group, \(\text{Fit}(G)\) its Fitting subgroup, \(\Phi(G)\) its Frattini subgroup, and \(\Phi_ f(G)\) the intersection of all maximal subgroups of finite index. Say \(G\) has property \(\nu\) when every non- nilpotent normal subgroup of \(G\) has a finite non-nilpotent \(G\)- quotient.
In this very readable paper the authors prove that \(G\) has property \(\nu\) if and only if \(\Phi_ f(G)\leq \text{Fit}(G)\), \(\text{Fit}(G)\) is nilpotent, and \(\text{Fit}(G/\Phi_ f(G))=\text{Fit}(G)/\Phi_ f(G)\). They also show that if \(G\) has finite rank then \(G\) has property \(\nu\) if and only if \(G\) is soluble-by-finite, \(\Phi(G)\leq \text{Fit}(G)\), \(\text{Fit}(G)\) is nilpotent, and \(\text{Fit}(G/\Phi(G))=\text{Fit}(G)/\Phi(G)\). Among the groups that are found to have property \(\nu\) are polycyclic-by-finite groups, free groups, subgroups of \(GL(n,\mathbb{Z})\), subgroups of finitely generated abelian-by-nilpotent-by-finite groups, and free metanilpotent groups.

MSC:

20E26 Residual properties and generalizations; residually finite groups
20F18 Nilpotent groups
20E28 Maximal subgroups
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups
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