How to describe this abelian group denoted by G
G=(Z+Z+Z)/N where Z denote the integers and + is direct sum and
N = <(7,8,9), (4,5,6), (1,2,3)> or the smallest submodule of Z+Z+Z containing these 3 vectors.
How would you describe G?
One could use a presentation to describe G which in this case is G = <a,b,c |7a+8b+9c=0, 4a+5b+6c=0, a+2b+3c=0> where a,b,c are the standard basis vectors for R^3.
However I do not know what G looks like in this form either.
One way to look at G is G=X/(q:X->X)
where X is R^3. q is the linear transformation given by the transformation matrix
7 4 1
8 5 2
9 6 3
But what does this form of G look like?
In other words I do not know what any of the 3 forms of G looks like.