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## Homework Statement

Clasify the group Z

_{4}xZ

_{2}/{0}xZ

_{2}using the fundamental theorem of finitely generated abelian groups.

## Homework Equations

FTOFGAG: In short it states that every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form Z

_{(p1)}

^{r1}x...xZ

_{(pn)}

^{rn}xZxZ...xZ

where pn's are prime numbers and rn's are +ve integers (p's,r's can be same)

Theorem 14.11 is just the fundamental theorem of homomorphism.

## The Attempt at a Solution

As given in the book

First time (at least from what I remember, and my memory span is that of a worm...) I encounter a projection map. But the main thing is, that I do not see how the theorem was applied ? I think I can justify to myself why pi(x,y) only gives x - otherwise, it would not be possible to make it isomorphic to Z

_{4}(I mean it would not make any sense to map Z

_{4}xZ

_{2}to only Z

_{4}), but why do we make such a choice in the first place ? And why can we say that {0}xZ

_{2}is the kernel ?