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

• Leb
In summary, the fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group is isomorphic to a direct product of cyclic groups. In this case, the projection map pi is used to show that Z4xZ2/{0}xZ2 is isomorphic to Z4, where {0}xZ2 is the kernel of pi. This is an example of the fundamental theorem of homomorphism.
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 ?

{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.

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

## 1. What is the group Z4xZ2/0xZ2?

The group Z4xZ2/0xZ2 is a quotient group, also known as a factor group, that is created by dividing the direct product of two groups, Z4 and Z2, by the trivial subgroup (0xZ2). This means that the elements of the group are cosets, or subsets, of the direct product that contain all the elements of the trivial subgroup.

## 2. How is the group Z4xZ2/0xZ2 classified?

The group Z4xZ2/0xZ2 is classified using the fundamental theorem of finitely generated abelian groups. This theorem states that any finitely generated abelian group can be expressed as a direct product of cyclic groups. Therefore, the group Z4xZ2/0xZ2 can be classified as a direct product of cyclic groups.

## 3. What does it mean for a group to be finitely generated?

A finitely generated group is a group that can be generated by a finite number of elements. This means that all the elements of the group can be written as combinations of a finite set of elements, known as generators. In the case of Z4xZ2/0xZ2, the group is generated by the elements (1,0) and (0,1).

## 4. How does the fundamental theorem of finitely generated abelian groups apply to Z4xZ2/0xZ2?

The fundamental theorem of finitely generated abelian groups states that any finitely generated abelian group can be expressed as a direct product of cyclic groups. In the case of Z4xZ2/0xZ2, the group is a direct product of two cyclic groups, Z4 and Z2. This means that the elements of the group can be written as combinations of the generators (1,0) and (0,1).

## 5. What is the significance of classifying a group using the fundamental theorem of finitely generated abelian groups?

Classifying a group using the fundamental theorem of finitely generated abelian groups allows us to understand the structure of the group and its elements. It also allows us to identify the generators of the group, which can help in solving problems and making calculations within the group. In the case of Z4xZ2/0xZ2, the classification allows us to express the elements of the group as combinations of the generators (1,0) and (0,1), making it easier to understand and work with.

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