AI 中文总结
本文研究群的局部可解根，利用Wilson序列证明局部（可解-有限）群等的相关根等式，给出根成员的共轭、换位子刻画及近模子群格群的对应结果。
AI 中文摘要
对于群G，令S(G)为满足对所有x∈G，⟨g,x⟩均可解的元素g∈G的集合。本文研究S(G)何时与局部可解根R_{L𝔖}(G)重合。利用Wilson的有限收敛字序列，证明对每个局部（可解-有限）群G及每个Wilson序列ω，有R_{L𝔖}(G)=R_{L}(G)=S(G)=W_ω(G)。同时得到根成员的四共轭、七换位子刻画，以及与6互素阶挠元的两共轭结果；该根等式对局部线性群成立，故当D为局部有限维除环时，对GL_∞(D)的子群也成立。此外独立证明，具有近模子群格的群满足R_{L𝔖}(G)=R_{L}(G)=S(G)。
英文摘要
For a group $G$, let $S(G)$ be the set of elements $g \in G$ such that $\langle g,x\rangle$ 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 $ω$, $$ R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G) = W_ω(G).$$ 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 $$R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G).$$
