Free Abelian Group Homomorphisms into S_3 - Count & Answer

[tgt]

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. ?

 

[yyat]

We can assume the free abelian group of rank 2 is $$\mathbb{Z}\times\mathbb{Z}$$.
The homomorphisms from $$\mathbb{Z}\times\mathbb{Z}$$ to $$S_3$$, correspond to pairs $$(a,b)\in S_3\times S_3$$ with $$ab=ba$$ via
$$f:\mathbb{Z}\times\mathbb{Z}\to S_3$$ corresponds to $$(f(1,0),f(0,1))$$.
(Show this!) So you just need to find all such pairs.

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

 

[tgt]

We can assume the free abelian group of rank 2 is $$\mathbb{Z}\times\mathbb{Z}$$.
The homomorphisms from $$\mathbb{Z}\times\mathbb{Z}$$ to $$S_3$$, correspond to pairs $$(a,b)\in S_3\times S_3$$ with $$ab=ba$$ via
$$f:\mathbb{Z}\times\mathbb{Z}\to S_3$$ corresponds to $$(f(1,0),f(0,1))$$.
(Show this!) So you just need to find all such pairs.

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

I understand your explanation. But there seem only to be 12 such pairs.

 

[yyat]

I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.

 

[tgt]

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.