From: Marston ConderResent-Date: Fri, 28 Jul 95 3:53:55 BST On Thu, 27 Jul 1995 Nigel Boston wrote >Does there exist a 3-generated finite group generated by a cyclic group >of order 2 and a non-abelian group of order 21 ? This is a specific case of >a question of Ribes and Wong, asked if, given finite groups H and K, there >exists a finite group G generated by groups isomorphic to H and K such that >d(G) = d(H) + d(K). The answer is Yes, and here is an example (of order 294): Let G be the semi-direct product of an elementary abelian group N = < a, b > of order 49 by a cyclic group Q = < c > of order 6, with c conjugating a to a^3, and b to b^3. Then b*c^3 generates a subgroup H of order 2, while a and c^2 generate a non-abelian subgroup K of order 21, and then since [b*c^3,c^2] = b^-1, together these subgroups generate G. Finally it's not difficult to show that G cannot be generated by two elements, because of the way Q acts on N [or because magma or gap will soon tell you that the "largest" 2-generator subgroups of G have orders 42 and 49]. Marston Conder