How can I show that the number of even permutations is equal to the number of odd permutations?
Yeah, that's precisely what CAN'T happen, it is actually phrased the exact way you did, with the appropriate negation insereted, in most textbooks. A different way of looking at this is that there is a natural action of on the set where and the action is . We can then think of the set of even actions as the stabilizer of from where everything I've told you follows.
If you haven't done group actions yet, just ignore that.
A finite set with two or more elements has equal numbers of even and odd permutations.
Proof. For each even permutation, we can obtain a unique odd permutation by transposing the first two elements. This defines a one-to-one correspondence between even and odd permutations, hence there are equal numbers of each.
I found this proof somewhere. Can you explain with an example what they mean by transposing the first two elements. This is what I was meaning...
the key fact here is that if a permutation can be expressed as a product of an odd (resp. even) number of transpositions, it can ONLY be expressed as a product of an odd (resp.even) number of transpositions.
this is a deep fact, and one which is often used to PROVE that [Sn:An] = 2. this is equivalent to proving the "signum" (or sign) function sgn:Sn-->{-1,1} is well-defined, after which it is easy to show that |An| = |Sn|/2, and that An is normal. one way of proving this is to invoke the Vandermonde polynomial:
and let a permutation σ in Sn act on p by taking:
odd permutations take p-->-p, while even permutations take p-->p.
my point is: all the difficulty is buried in ensuring that "even permutation" and "odd permutation" are well-defined terms, that we cannot have a permutation that is both. until this is done, An is not a well-defined set.
if we call our initial (even) permutation σ, and our second permutation τ, then this means:
τ(1) = σ(2)
τ(2) = σ(1)
τ(k) = σ(k), k = 3,4,...,n
this is equivalent to writing: τ = (σ(1) σ(2))σ.
note that τ has one more transpostion in its product than σ does.