Question 14132
 The decomposition of the given group G is called direct product if
 G = H x K for subgroups H and K of G.

 In case of abelian, G is called the direct sum of H & K and denoted
 by H + K.

 The example that you gave is merely OK, since the form you wrote was
 not clear and not in good shapes.
  
 Just close your eyes, there are thousands of such examples as
 R^2 = R + R, R^3 = R^2 + R. 

 Your example should express what your addition group G of  2x2 matrices over Z,
 over Q or R or C (whatever)
  If G is the addition group of 2x2 matrices over Z.
 Since dim G = 4, let H = {[a b]
                            [0 0] | a,b in Z}
 and K = {[0 0]
          [c d] | c,d in Z}

 then G = H x K [Note dim H = dim K= 2]

 For finite group , let G = Z6 (i.e Z6 = Z/6Z,mod group of Z )
 H = {[0],[2],[4]} , K = {[0],[3]}
 then G = H x K. (why ?)

 In general, Zmn is isomorphic to Zm x Zn if m & n are relative prime.

 Try to look for more examples in the web or books.

 Of course, you have to work hard.

 Kenny