here the group elements of the resulting group are sets containing an element from each on the multiplicands.
Example 1
Lets take the simplest example we can think of. Lets take the group Z_{2} which is the integers modulo two which gives an exclusive or table:
Cayley Table 
Cayley Graph 


Lets combine two of these groups to give: Z_{2}Z_{2}
{g, h} × {g' , h' } = {g * g' , h o h' }
where:
 × is the operation of the combined algebra.
 * is the operation of the group G.
 o is the operation of the group H which may be, or may not be, the same as *.
Cayley Table 


Cayley Graph 
Is there any way we can modify this to generate the complex numbers?
Can we then go on to generate the quaternions and octonions?
 H = CC
 O = HH