Invariant generation (Q750609)

Definition. A (profinite) group G is invariantly generated by its (closed) subgroups \(G_ 0,...,G_ n\) if, for any \(g_ 0,...,g_ n\in G\), one has \(G=<G_ 0^{g_ 0},...,G_ n^{g_ n}>\). Main result: Let \(G_ 1\) and \(G_ 2\) be nontrivial profinite groups. Then their free product is not invariantly generated by \(G_ 1\) and \(G_ 2\).