Why are Even and Odd permutations well-defined? [duplicate]

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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?

Why are Even and Odd permutations well-defined?

Thank You

marked as duplicate by José Carlos Santos, YuiTo Cheng, Dietrich Burde abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 16 at 11:53

• – Martin R Jun 16 at 11:47
• How do you define an even and an odd permutation? – Dietrich Burde Jun 16 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 at 11:52
• See Cartier's paper e-periodica.ch/cntmng?pid=ens-001:1970:16::12 – Lord Shark the Unknown Jun 16 at 11:56

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.