Why are Even and Odd permutations well-defined?

Question

I'm currently studying for a theory exam in Abstract Algebra. One question is about first defining the concepts of even and odd permutations and then explain why these concepts are well-defined. So, I find this a bit difficult and I'm wondering if you guys can help me out?

Thank You

Related (if not duplicate): Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions — Martin R, Jun 16 '19 at 11:47
To start off, you can write out the definition of even and odd permutations you're working with, and then tell us which part of the definition requires additional explanation. — tia, Jun 16 '19 at 11:52
See Cartier's paper https://www.e-periodica.ch/cntmng?pid=ens-001:1970:16::12 — Angina Seng, Jun 16 '19 at 11:56

score 1 · Answer 1 · answered Jun 16 '19 at 11:51

The definitions of even and odd that I'm familiar with have to do with expressing a permutation as a product of transpositions.

Specifically, every permutation $\sigma$ can be expressed as a product of transpositions $\tau_1 \tau_2 \cdots \tau_n$. Then $\sigma$ is even if $n$ is even, and $\sigma$ is odd if $n$ is odd, where $n$ is the number of transpositions appearing in the product.

The problem is that there are lots of ways of expressing a permutation as a product of transpositions. For example $$(1~~2~~3) = (1~~3)(1~~2) = (2~~1)(2~~3) = (1~~2)(1~~3)(2~~3)(2~~1)$$ So you need to show that an even permutation $\sigma$ doesn't magically become odd just by expressing it in a different way as a product of transpositions—that is, you need to show that if some expression of $\sigma$ as a product of transpositions has an even (resp. odd) number of transpositions in the product, then all such expressions do.

Ok! This is good. Thanks! – Victor Galeano Jun 17 '19 at 08:12 — Victor Galeano, Jun 17 '19 at 08:12

Why are Even and Odd permutations well-defined?

1 Answers1