From: Marston Conder 
Resent-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