Question

5. [2 marks] Write down the relation matrix of the abelian group Now reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups.

Answer 1

The first relation can be written as $\begin{align*}3x+3y+4z-w=0\end{align*}$

Therefore, the relation matrix is

4604 5705 334

Performing elementary row and column operations, we get

4 0 0 3 5 4 3 76 4 0 0 0 22 18 4 0 0 0 22 18 0 20 16 0 22 18 (Ка Ка-3R1) 0 20 16 0 20 16

1 0-4 0 22 18 0 20 16 0 20 16 0 22 18 0 20 16 0 20 16 0 4 18 0 4 16 0 4 16 0 0 4

which is the Smith Normal Form. Thus, are the diagonal entries, showing that

0,1,2,4

$\begin{align*}G&\cong {\mathbb Z}\oplus {\mathbb Z}/{\mathbb Z}\oplus {\mathbb Z}/2{\mathbb Z}\oplus {\mathbb Z}/4{\mathbb Z}\\ &\cong {\mathbb Z}\oplus {\mathbb Z}/2{\mathbb Z}\oplus {\mathbb Z}/4{\mathbb Z}\end{align*}$

which is a direct sum of cyclic groups.