On Pro-$p$ Cappitt Groups with finite exponent

Abstract

A pro-$p$ Cappitt group is a pro-$p$ group $G$ such that $\tilde{S}(G) = \overline{ \langle L \leqslant _c G \:|\, L \ntriangleleft G \rangle }$ is a proper subgroup (i.e. $\tilde{S}(G) \ne G$). In this paper we prove that non-abelian pro-$p$ Cappitt groups whose torsion subgroup is closed and it has finite exponent. This result is a natural continuation of main result of the first author [7]. We also prove that in a pro-$p$ Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show that pro-$2$ Cappitt groups of exponent $4$ are pro-$2$ Dedekind groups. These results are pro-$p$ versions of the generalized Dedekind groups studied by Cappitt (see Theorem 1 and Lemma 7 in [1]).

Keywords

• Generalized Dedekind groups
• pro-$p$ Cappitt groups
• torsion groups.