2

Consider the parity homomorphism of the symmetric group $$ p:S_n\to Z/(2). $$ Is it possible to characterise this map by a pure universal property?

This question occurred to me when I was reading this post Is there a "natural" / "categorical" definition of the "parity" of a permutation?

Community
  • 1
Ma Ming
  • 7,522

1 Answers1

7

Yes. It's the abelianization; that is, it's universal with respect to maps from $S_n$ into abelian groups.

Qiaochu Yuan
  • 468,795
  • Note that there is no issue with identifying the abelianization with $\mathbb{Z}/2\mathbb{Z}$; this identification requires no choices because $\mathbb{Z}/2\mathbb{Z}$ has no nontrivial automorphisms. It is in fact the unique nontrivial group with this property. – Qiaochu Yuan Apr 26 '14 at 05:32
  • We need $n \geq 2$ (otherwise $S_n$ and hence its abelianization is trivial). – Martin Brandenburg Apr 27 '14 at 07:27
  • @Martin: but in that case so is the parity map. That seems fine to me. – Qiaochu Yuan Apr 27 '14 at 17:37
  • No, the abelianization is surjective. – Martin Brandenburg Apr 27 '14 at 18:28
  • @Martin: I don't understand why you're arguing this. It doesn't seem to make any conceivable difference to me whether the parity map for $S_1$ is regarded as a map $S_1 \to S_1/[S_1, S_1]$ or as a map $S_1 \to \mathbb{Z}/2\mathbb{Z}$. – Qiaochu Yuan Apr 27 '14 at 21:46
  • The codomains are different (hence the morphisms differ). – Martin Brandenburg Apr 27 '14 at 22:03
  • @Martin: okay, sure, obviously I know that, but is that difference important for anything? I can't think of any reason why I would really want parities for $S_1$ to take values in $\mathbb{Z}/2\mathbb{Z}$. – Qiaochu Yuan Apr 27 '14 at 23:30