Locally Solvable Radicals via Wilson Radical Sets and Subgroup Lattices

Abstract

For a group $G$, let $S(G)$ be the set of elements $g\in G$ such that $⟨g,x⟩$ is solvable for all $x\in G$. We study when $S(G)$ coincides with the locally solvable radical $R_{\mathrm{L}\mathfrak{S}}(G)$. Using Wilson's profinitely convergent word sequences, we show that, for every locally (solvable-by-finite) group $G$ and every Wilson sequence $\omega$, $$R_{\mathrm{L}\mathfrak{S}}(G)=R_{\mathrm{L}\mathfrak{R}}(G)=S(G)=W_{\omega }(G).$$ We also obtain four-conjugate and seven-commutator descriptions of radical membership, together with a two-conjugate result for torsion elements of order coprime to $6$. The same radical identity holds for locally linear groups, and hence for subgroups of ${\mathrm{GL}}_{\infty }(D)$ when $D$ is a locally finite-dimensional division ring. Independently, we prove that groups with nearly modular subgroup lattice satisfy $$R_{\mathrm{L}\mathfrak{S}}(G)=R_{\mathrm{L}\mathfrak{R}}(G)=S(G).$$