Groups that involve finitely many primes and have all subgroups subnormal. II. (Q2902676)
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scientific article; zbMATH DE number 6069859
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Groups that involve finitely many primes and have all subgroups subnormal. II. |
scientific article; zbMATH DE number 6069859 |
Statements
22 August 2012
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subnormal subgroups
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residually nilpotent groups
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nilpotency
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hypercentral groups
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torsion subgroup
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Groups that involve finitely many primes and have all subgroups subnormal. II. (English)
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\textit{C. Casolo} [Rend. Circ. Mat. Palermo (2) 50, No. 2, 321-324 (2001; Zbl 1138.20306)] and \textit{H. Smith} [Arch. Math. 76, No. 1, 1-6 (2001; Zbl 0982.20018)] have shown that a torsion-free group with all subgroups subnormal is nilpotent and in part I [\textit{H. Smith}, J. Algebra 347, No. 1, 133-142 (2011; Zbl 1253.20023)] it is shown that if \(G\) is a group with all subgroups subnormal and if the torsion subgroup of \(G\) involves just finitely many primes then \(G\) is nilpotent-by-divisible Chernikov.NEWLINENEWLINE In this paper further results are obtained concerning groups with all subgroups subnormal in which the torsion subgroup is a \(\pi\)-group for some finite set of primes \(\pi\). For example, a hypercentral or residually nilpotent group satisfying these hypotheses is nilpotent.
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