On varieties where $\mathrm{CS}\mathfrak{X}$ implies $\mathfrak{X}\mathrm{T}$

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The article provides new examples of varieties where the property $\mathrm{CS}\mathfrak{X}$ (all maximal $\mathfrak{X}$-subgroups are malnormal) implies $\mathfrak{X}\mathrm{T}$ (any two $\mathfrak{X}$-subgroups with a nontrivial intersection generate an $\mathfrak{X}$-subgroup), extending the findings from previous work. This implication matters as it clarifies the relationship between these group properties within specific varieties, contributing to the broader understanding of $\CSX$- and $\XT$-groups. One concrete detail is that while $\mathrm{CS}\mathfrak{X}$ does not generally imply $\mathfrak{X}\mathrm{T}$, certain identified varieties indeed satisfy this implication.

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Checking out a new math paper on varieties where certain group properties imply each other. Interesting!

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arXiv:2606.08007v2 Announce Type: replace Abstract: In our previous work \cite{Omar-Shah2}, we initiated the study of $\CSX$- and $\XT$-groups associated with a fixed variety $\X$. A group belongs to the former class if all of its maximal $\X$-subgroups are malnormal, and to the latter if any two $\X$-subgroups with nontrivial intersection generate an $\X$-subgroup. In general, $\CSX$ does not imply $\XT$, however as shown in \cite{Omar-Shah2}, some varieties do satisfy this implication. In this article, we provide additional examples of varieties for which $\CSX$ implies $\XT$.