Why is the parity of a permutation an important concept? In Pinter's A Book Of Abstract Algebra, the author states that:

A number of great theorems of mathematics depend for their proof (at that crucial step when the razor of logic makes its decisive cut) on none other than the distinction between even and odd permutations.

What are these "great theorems"? Where do permutation parities manifest themselves in concrete mathematics? The only place where I've seen parity show up naturally is in the signs for the various terms of a determinant. There's also that 15 puzzle thing.
Alternating groups are important in group theory because they're simple, so I'd like to get a firmer intuitive grasp on them. Part of that involves seeing them show up in practice.
 A: I do not know what those great theorems are about, but I thought I can still give my thoughts on the subject. 
I've always understood odd/evenness for permutations in the same way I understand them for numbers, it is a fundamental way to distinguish them, i.e. it is a fundamental property of a given permutation, a way to divide them into two parts.
Now this always allows us to get a one-dimensional representation (the alternating rep) other than the trivial one for a group that is dividable as such. E.g. for the cycle group $\left\{ a, a^2, a^3,\ldots, a^p\right\}$ we can get the alternating rep. if we send group elements with uneven powers to -1 and elements with even powers to +1. In the same way we can find
an alternating representation for $S_n$ using odd/evenness for permutations. All you need is a way to split up your group in a way that satisfies the 'minus*minus=plus etc.' sort of system.
Now where does this come up naturally. Only example I can come up of the top off my head is that fermions transform under the alternating representation of the symmetric group $S_n$. I.e. switchting two fermions gives a -1, which has huge consequences on the statistics of those particles (compared to bosons).
Also, regarding your puzzle 15 link. It's also important in solving the Rubik's cube. If you have the wrong number of permutations of certain flips, the Rubik's cube is no longer solvable.
Sorry, I can't really answer your question, but at least it is a maybe a start/ a way to think about parity.
