TITLE: Intuitive explanation of even/odd permutation
QUESTION [9 upvotes]: Given a permutation it can be classified as either even or odd depending on whether it is expressible as a product of even or odd number of transpositions. Is there some geometrical or intuitive meaning for this? Secondly what is the significance of classifying permutations as even or odd?
Thanks
Update: I am looking for a motivation for the classification of permutations as even and odd when a student first encounters them. For example rotations in $D_n$ can be used to motivate $\mathbb{Z}_n$. Likewise I was looking for a general motivation for even/odd permutations. At an advanced level there is the motivation that $A_n$ is simple for $n\ge 5$, but is there any motivation for a novice?

REPLY [2 votes]: If you consider orientation in Euclidean space to be intuitive, then the answer is affirmative: a permutation on $n$ symbols is even or odd according to whether it preserves or reverses orientation when acting on the standard basis of $n$-space.