Classify the group Z4xZ2/0xZ2 using fund.thm. of finetely gen. abl. grps.

[Leb]

Homework Statement

Clasify the group Z₄xZ₂/{0}xZ₂ 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)^r1x...xZ_(pn)^rnxZxZ...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 Z₄ (I mean it would not make any sense to map Z₄xZ₂ to only Z₄), but why do we make such a choice in the first place ? And why can we say that {0}xZ₂ is the kernel ?

 

[sunjin09]

{0}xZ2 is the kernel of pi because pi(0xz2)=0 for any z2 in Z2. It can be shown that pi: Z4xZ2→Z4 is a homomorphism, therefore the canonical map Pi: Z4xZ2/{0}xZ2→Z4 is isomorphism, so Z4xZ2/{0}xZ2 and Z4 are isomorphic. I don't see why you need the theorem about finite abelian group, maybe this is just an example of that theorem.

 

[Leb]

Thanks sunjin09!
So it is all about how we define pi.