# Structure theorem

• Dec 16th 2009, 10:24 AM
james123
Structure theorem
I'm not sure how to begin the second part of the following problem.

Quote:

Originally Posted by The problem
If $G=\mathbb{Z}_8\times \mathbb{Z}_{12} \times \mathbb{Z} \times \mathbb{Z}$ and H is the subgroup of G generated by (4,8,1,0) and (1,1,2,3),
(i) find the decomposition of H as the direct sum of cyclic groups of prime power orders and a free abelian group;
(ii) do the same for G/H.

For part (i) I just set up a 4x4 matrix with rows (8, 0, 0, 0), (0, 12, 0, 0) (which take into account of G) and (4, 8, 1, 0), (1, 1, 2, 3) (corresponding to H), row reduced over the integers to get a diagonal matrix with entries 1, 1, 4, 12996, so $H\approx \{0\}\times\{0\}\times \mathbb{Z}_4\times \mathbb{Z}_{12996} \approx \mathbb{Z}_{2^2}\times \mathbb{Z}_{2^2} \times \mathbb{Z}_{3^2} \times \mathbb{Z}_{19^2}$. Their accuracy is irrelevent to my question: I would just appreciate a hint as to how (ii) can be attacked using (i).