The problem involves determining the number of different homomorphisms from a free abelian group of rank 2, specifically \(\mathbb{Z} \times \mathbb{Z}\), into the symmetric group \(S_3\). Participants are exploring the relationship between elements of the group and the structure of \(S_3\).
The discussion is active, with participants presenting different counts of homomorphisms and clarifying their reasoning. Some guidance has been offered regarding the types of elements in \(S_3\) and their implications for counting pairs. There is no explicit consensus on the correct number of homomorphisms, but productive dialogue is ongoing.
Participants are working under the assumption that the free abelian group is \(\mathbb{Z} \times \mathbb{Z}\) and are considering the implications of this structure on the homomorphisms into \(S_3\). There is a noted discrepancy in the counts provided by different participants, which may stem from differing interpretations of what constitutes distinct pairs.
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.
yyat said:I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.