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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?

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

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