The local embedding of the Hall group into the class of finite groups (Q5947531)

The following definition is due to \textit{A. M. Vershik} and \textit{E. I. Gordon} [see Algebra Anal. 9, No. 1, 71-97 (1997; Zbl 0898.20016)]. A group \(G\) is called locally embedded into finite groups (LEF) if for any finite subset \(S\subset G\) there exist a finite group \(K\) with multiplication \(a\cdot_Kb\in K\), such that \(S\subset K\subset G\) and for any \(a,b\in S\) \(a\cdot_Kb=ab\) if \(ab\in S\). For example, any residually finite group is an LEF group. The author informs that there exists Kharlampovich's example of a finitely generated 3-soluble group that is not LEF. A. M. Vershik asked whether \textit{P. Hall}'s 3-soluble not residually finite group [Proc. Lond. Math. Soc., III. Ser. 9, 595-622 (1959; Zbl 0091.02501)] is also such a group. It is proved that Hall's group is LEF.