Free Abelian Group Homomorphisms into S_3 - Count & Answer

In summary, there are 18 different homomorphisms of a free abelian group of rank 2 into S_{3}. This is determined by finding all possible pairs of elements in S_3 that satisfy the condition ab=ba, with a corresponding to f(1,0) and b corresponding to f(0,1). There are only three types of elements in S_3: the identity, the transpositions, and the cyclic permutations, which can be used as a guide in enumerating these pairs. It is important to note that pairs (a,b) and (b,a) cannot be counted as the same, as they result in different homomorphisms.
  • #1
tgt
522
2

Homework Statement


How many different homomorphisms are there of a free abelian group of rank 2 into S_{3}?

Where S_{3} is the symmetric group of 3 elements.

The Attempt at a Solution


I think 12 but the answers suggest 18. ?
 
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  • #2
We can assume the free abelian group of rank 2 is [tex]\mathbb{Z}\times\mathbb{Z}[/tex].
The homomorphisms from [tex]\mathbb{Z}\times\mathbb{Z}[/tex] to [tex]S_3[/tex], correspond to pairs [tex](a,b)\in S_3\times S_3[/tex] with [tex]ab=ba[/tex] via
[tex]f:\mathbb{Z}\times\mathbb{Z}\to S_3[/tex] corresponds to [tex](f(1,0),f(0,1))[/tex].
(Show this!) So you just need to find all such pairs.

Tip for enumerating these pairs: There are only three types of elements in [tex]S_3[/tex]: the identity, the transpositions and the cyclic permutations.
 
Last edited:
  • #3
yyat said:
We can assume the free abelian group of rank 2 is [tex]\mathbb{Z}\times\mathbb{Z}[/tex].
The homomorphisms from [tex]\mathbb{Z}\times\mathbb{Z}[/tex] to [tex]S_3[/tex], correspond to pairs [tex](a,b)\in S_3\times S_3[/tex] with [tex]ab=ba[/tex] via
[tex]f:\mathbb{Z}\times\mathbb{Z}\to S_3[/tex] corresponds to [tex](f(1,0),f(0,1))[/tex].
(Show this!) So you just need to find all such pairs.

Tip for enumerating these pairs: There are only three types of elements in [tex]S_3[/tex]: the identity, the transpositions and the cyclic permutations.

I understand your explanation. But there seem only to be 12 such pairs.
 
  • #4
I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.
 
  • #5
yyat said:
I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.

I see now. thank you.
 

1. What is a free abelian group homomorphism into S_3?

A free abelian group homomorphism into S_3 is a function that maps elements from a free abelian group to elements in the symmetric group S_3 while preserving the group structure. In other words, the function must map the identity element, preserve group operations (multiplication and inverse), and maintain the commutativity property.

2. How do I count the number of free abelian group homomorphisms into S_3?

To count the number of free abelian group homomorphisms into S_3, you can consider the number of generators in the free abelian group and the number of generators in S_3. The number of homomorphisms will be equal to the number of possible mappings between the generators.

3. What are some examples of free abelian group homomorphisms into S_3?

An example of a free abelian group homomorphism into S_3 is the trivial homomorphism, which maps all elements in the free abelian group to the identity element in S_3. Another example is the canonical homomorphism, which maps each generator in the free abelian group to a specific permutation in S_3.

4. How are free abelian group homomorphisms into S_3 used in mathematics?

Free abelian group homomorphisms into S_3 are used in various areas of mathematics, such as group theory, abstract algebra, and topology. They can be used to study the structure and properties of free abelian groups and symmetric groups, and to prove theorems and solve problems in these fields.

5. Are there any restrictions on the elements that can be mapped by a free abelian group homomorphism into S_3?

Yes, there are restrictions on the elements that can be mapped by a free abelian group homomorphism into S_3. The elements must be from a free abelian group and must satisfy the homomorphism property, which means they must preserve the group structure. Additionally, the elements must be compatible with the structure of S_3 and maintain the commutativity property.

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