On the minimal number of generators of free profinite products of profinite groups (Q2705009)

An open question known to specialists for a long time is settled in this paper; namely the question of the validity of the Grushko-Neumann theorem in the category of profinite groups. The Grushko-Neumann theorem states that the minimal number of generators \(d(G)\) of a free product \(G=G_1*G_2\) is equal to \(d(G_1)+d(G_2)\). In the profinite context one thinks in terms of topological generators. The author proves that a profinite version of the Grushko-Neumann theorem does not hold.NEWLINENEWLINENEWLINEThe main result of the paper can be reformulated as follows: Theorem. Let \(H_1,H_2\) be profinite groups of coprime orders and \(G=H_1\amalg H_2\) their free profinite product. Then for any positive integer \(e\) there exists an integer \(\overline d\) such that for any \(d\geq\overline d\) the following is true: if \(H_1\) is generated by \(d\) involutions and \(d(H_2)\leq d\) then \(d(G)\leq 2d-e\).NEWLINENEWLINENEWLINEThe author formulates the theorem in terms of finite groups and methods of finite group theory are used in the proof. In particular, the proof depends on the classification of finite simple groups.