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Free abelian group?

  • Thread starter tgt
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tgt

520
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1. 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.

3. The Attempt at a Solution
I think 12 but the answers suggest 18. ?????
 
316
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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:

tgt

520
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.
I understand your explanation. But there seem only to be 12 such pairs.
 
316
0
I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.
 

tgt

520
2
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.
 

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