On the Number of Generators of a Free Product, and a Lemma of Alexander Kurosch

Let a group G be generated by elements a₁,..., a_m which are supposed to be ≠I, but not by a smaller number of elements of G, then this minimum number of generators will be denoted by

m(G).                                                           (I)

If the order of G is equal to one, put m(G) = 0. The invariant m(G) of a group G has not yet been much investigated. One may suppose that the minimum number of generators of a free product is equal to the sum of the corresponding numbers of the free factors, but this supposition (which is true for large classes of groups) has neither been confirmed by a general proof, nor been refuted by a counter-example.