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.