# Homomorphisms from a free abelian group

• ehrenfest
In summary, the conversation discusses the number of homomorphisms of a free abelian group of rank 2 into two different groups, Z_6 and S_3. The upper bound for both is 36. The proof for part a involves defining a homomorphism by assigning values to the generators of the free abelian group and using the fact that Z_6 is abelian. The answer for part b is 18, as the quotient group of an abelian group must also be abelian.
ehrenfest

## Homework Statement

How many homomorphism are there of a free abelian group of rank 2 into a) Z_6 and b) S_3.

## The Attempt at a Solution

Since the images of the generators completely determine a homomorphism, the upper bound for both is 36.
Now a free abelian group of rank 2 is isomorphic to Z \times Z, which has basis {(1,0),(0,1)} and by the definition of a free abelian group, every nonzero element in the group can be uniquely expressed in the form a(1,0)+b(0,1) with a,b in Z. Therefore, given two arbitrary elements x and y in Z_6, we define phi((0,1)) = x and phi((1,0)) = y and for an arbitrary nonzero c = a(1,0)+b(0,1) in Z \times Z, we define $\phi(c) = a\cdot x+b\cdot y$. And then define \phi(0) = 0.

Now, why is that a homomorphism? Because if d and e are arbitrary nonzero elements in Z \times Z, such that d = a_1(1,0)+b_1(0,1) and e = a_2(1,0)+b_2(0,1), then we have
$$\phi(d+e) = \phi((a_1+a_2)(1,0)+(b_1+b_2)(0,1)) = (a_1+a_2)x+(b_1+b_2)y = (a_1x+b_1y)+(a_2x+b_2y) = \phi(d)+\phi(e)$$
where in the second-to-last step I have used the fact that Z_6 is abelian. If d is 0 then we have $\phi(d+e)=\phi(e)=\phi(e)+0=\phi(e)+\phi(0)$.

Therefore the answer to part a is 36. Please confirm that proof. I have absolutely no idea how to do part b since that group is not abelian.

anyone?

I quickly scanned your proof, and it looks alright. Although you can just use the universal property for free groups: if you send (1,0) and (0,1) to G, then this extends to a unique homomorphism of ZxZ into G. This is true for all free groups, not just abelian ones. So part (b) follows similarly.

I am kind of confused about your last post. Please read the problem again. The answer to part b is 18 not 36 which means what you said cannot be true.

Forget my last post. I'm not sure what was going through my head at the time.

If f:ZxZ->S_3 is a homomorphism, then (ZxZ)/kerf =~ imf. What does this tell us?

Last edited:
I am not sure what you are getting at. Oh wait--any quotient group of an abelian group must be abelian so the image of f must be abelian. Is that what you meant?

## What is a free abelian group?

A free abelian group is a mathematical structure that consists of a set of elements and a binary operation, typically denoted by addition, that satisfies certain properties. In a free abelian group, the elements can be combined in any way without any restrictions or constraints.

## What is a homomorphism?

A homomorphism is a mathematical function that preserves the structure of a group. In other words, it maps elements of one group to elements of another group in a way that respects the group operations. In the context of free abelian groups, a homomorphism is a function that preserves the addition operation.

## How are homomorphisms from a free abelian group different from other types of homomorphisms?

Homomorphisms from a free abelian group are different from other types of homomorphisms because they preserve the additive structure of the group. This means that the function only needs to preserve the addition operation, whereas other types of homomorphisms may need to preserve multiple operations, such as multiplication or division.

## What is the importance of studying homomorphisms from a free abelian group?

Studying homomorphisms from a free abelian group is important in many areas of mathematics, including abstract algebra, number theory, and topology. These homomorphisms can be used to understand the structure of free abelian groups and to solve problems involving these groups.

## How are homomorphisms from a free abelian group used in real-world applications?

Homomorphisms from a free abelian group have many applications in computer science, physics, and engineering. They are used in coding theory, cryptography, and signal processing, among other areas. They also have important applications in the study of symmetry and symmetry breaking in physical systems.

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