On pro-\(p\) Cappitt groups with finite exponent. (English) Zbl 1537.20070
Let \(G\) be a group and let \(S(G)=\langle N \mid N < G, N \ntrianglelefteq G \rangle\) be the subgroup of \(G\) generated by all proper subgroups which are not normal in \(G\). It is clear that \(S(G)\) is a characteristic subgroup of \(G\) and \(G/S(G)\) is a Dedekind group. A group \(G\) is a Cappitt group if \( S(G) \not =G\) (see [D. Cappitt, J. Algebra 17, 310–316 (1971; Zbl 0232.20067)]). In the profinite version, a profinite Dedekind group will be a profinite group in which every closed subgroup is normal. A pro-\(p\) Cappitt group is a pro-\(p\) group \(G\) such that \(\widetilde{S}(G)=\overline{\langle N \mid N <_{c} G, N \ntrianglelefteq G \rangle} \not =G\).
In the paper under review, the authors prove Theorem A: Let \(G\) be a non-abelian pro-\(p\) Cappitt group. Then \(G'\) is a procyclic central subgroup. Moreover, if \(\mathrm{tor}(G) \leq_{c}, G\) then \(G\) has finite exponent. This result is a natural continuation of the main result of the first author (see [Quaest. Math. 44, No. 3, 307–311 (2021; Zbl 1475.20045)]). They also prove that pro-\(2\) Cappitt groups of exponent \(4\) are pro-\(2\) Dedekind groups.
In the paper under review, the authors prove Theorem A: Let \(G\) be a non-abelian pro-\(p\) Cappitt group. Then \(G'\) is a procyclic central subgroup. Moreover, if \(\mathrm{tor}(G) \leq_{c}, G\) then \(G\) has finite exponent. This result is a natural continuation of the main result of the first author (see [Quaest. Math. 44, No. 3, 307–311 (2021; Zbl 1475.20045)]). They also prove that pro-\(2\) Cappitt groups of exponent \(4\) are pro-\(2\) Dedekind groups.
Reviewer: Egle Bettio (Venezia)
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GAPReferences:
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