What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set X preserves no structure: or, in other words, it preserves only the structure of being a set. When X is finite, what structure can the alternating group be said to preserve?

As a way of making the question precise, is there a natural definition of a category C equipped with a faithful functor to FinSet such that the skeleton of the underlying groupoid of C is the groupoid with objects Xn such that Aut(Xn)An?

Edit: I’ve been looking for a purely combinatorial answer, but upon reflection a geometric answer might be more appropriate. If someone can provide a convincing argument why a geometric answer is more natural than a combinatorial answer I will be happy to accept that answer (or Omar’s answer).

Answer

The alternating group preserves orientation, more or less by definition. I guess you can take C to be the category of simplices together with an orientation. I.e., the objects of C are affinely independent sets of points in some Rn together with an orientation and the morphisms are affine transformations taking the vertices of one simplex to the vertices of another. Of course this is cheating since if you actually try to define orientation you’ll probably wind up with something like “coset of the alternating group” as the definition. On the other hand, some people find orientations of simplices to be a geometric concept, so this might conceivably be reasonable to you.

Attribution
Source : Link , Question Author : Qiaochu Yuan , Answer Author : Michael Hardy

Leave a Comment