# Classification of homomorphism as abelian group

Consider $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}^4$ $$f(a, b, c) = (a+2b+8c, 2a-2b+4c, -2b+12c, 2a -4b + 4c)$$

Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of $\mathbb{Z}^4$ by the image of this homomorphism as an abstract abelian group.

I first put the kernel into Smith normal form which has entries 1, 2, 12.

I think that this means that the image is $C_2 X C_{12}$? How do I do the last part of the question?

• what book is this from? – Will Jagy Jun 8 '18 at 0:45
• It's not from a book, it's the start of a uni exam question – KingJ Jun 8 '18 at 11:09
• Show us your work. – lhf Jun 8 '18 at 12:34