Closed normal subgroups of free pro-\(S\)-groups of finite rank. (Q2882811)

Fix a non-Abelian finite simple group \(S\) and let \(\mathcal S\) denote the class of poly-\(S\) groups, i.e.\ the class of finite groups which have all composition factors isomorphic to \(S\). A pro-\(\mathcal S\) group is an inverse limit of groups in \(\mathcal S\). Denoting by \(M(G)\) the intersection of all maximal open normal subgroups of a pro-\(\mathcal S\) group \(G\), the \(S\)-rank of \(G\) is the cardinal \(\alpha\) such that \(G/M(G)\cong S^\alpha\). Results of \textit{O.~V.~Mel'nikov} [Math. USSR, Izv. 12, 1-20 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 3-25 (1978; Zbl 0382.20032)] on free pro-\(\mathcal S\) groups of infinite rank lead to the following problem: does a free pro-\(\mathcal S\) group of given finite rank \(e\) admit for each \(n\in\mathbb N\) a closed normal subgroup of \(\mathcal S\)-rank \(n\)?NEWLINENEWLINE In [J. Group Theory 13, No. 5, 759-767 (2010; Zbl 1211.20025)], \textit{L.~Fireman} showed that the problem reduces to the following question about poly-\(S\) groups: given \(e\geq 2\), does there exist for each \(n\in\mathbb N\) an \(e\)-generated poly-\(S\) group with normal subgroup isomorphic to \(S^n\). The author of the present short note provides the following, almost complete answer. Let \(l(S)\) denote the minimum among the indices of proper subgroups of \(S\). Then (i) for every \(n\in\mathbb N\) there exists a \(3\)-generated poly-\(S\) group with normal subgroup isomorphic to \(S^n\) and (ii) if \(S^{l(S)}\) is \(2\)-generated then for every \(n\in\mathbb N\) there exists a \(2\)-generated poly-\(S\) group with normal subgroup isomorphic to \(S^n\). It is known that the condition in (ii) holds whenever \(S\) has sufficiently large order. Moreover, it is explained that one cannot expect to weaken the condition very much.NEWLINENEWLINE The construction of poly-\(S\) groups \(G\) of the desired form involves \(k\)-fold iterated wreath products \(L(S,k)=S\wr\cdots\wr S\), formed with respect to a fixed transitive permutation representation of \(S\) in degree \(l(S)\). Each \(L=L(S,k)\) is a monolithic group and bounds for the minimal number of generators of crown-based powers \(L_t\) of \(L\) yield the necessary estimates for the minimal number of generators of the manufactured groups \(G\). The construction implicitly relies on facts from the classification of finite simple groups, for instance, that the group \(S\) is \(2\)-generated.