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  • 11
    $\begingroup$ I think this one is "the" answer. $\endgroup$ Commented Mar 9, 2022 at 14:48
  • 55
    $\begingroup$ This has been an educational MO question for me, but I feel that I've learned more about mathematicians than about mathematics. Clearly, some other people's intuitions don't line up with my own, and I feel like I have to reverse engineer the mindset that makes certain arguments seem natural and others seem contrived. $\endgroup$ Commented Mar 10, 2022 at 17:18
  • 17
    $\begingroup$ @HJRW Yes, I don't find it more natural. Roughly speaking, it gives an algebraic proof of what I consider to be a geometric fact. Nothing wrong with that, but I don't find it any more explanatory than, say, simply counting inversions. That $A_n$ is the group of rotations of a simplex is, to me, the explanation for its existence. $\endgroup$ Commented Mar 11, 2022 at 13:58
  • 20
    $\begingroup$ Even in 3 dimensions, capturing our geometric intuition of a rotation is formally difficult. The conventional approach requires us to construct the real numbers and describe continuous transformations that preserve the metric. But to me this does not imply that rotations in 3 dimensions are a complicated concept. It says more about the gap between our intuitive grasp of geometry and our ability to formalize it. $\endgroup$ Commented Mar 11, 2022 at 17:06
  • 10
    $\begingroup$ For the sake of others: To say that $D$ is a torsor under $\{\pm1\}^E$ just means that $\{\pm 1\}^E$ acts simply transitively on $D$. The construction of $Sym(X) \to Sym(D/G)$ is conceptual in the sense asked for, that no auxiliary computation is required to verify that it is well-defined or that it is a homomorphism. To me, it is obvious that the construction $X \mapsto D/G$ defines a functor from the category (finite sets of size $\ge 2$, bijections) to the category (size $2$ sets, bijections), and then it is automatic that one gets $Sym(X) \to Sym(D/G)$. $\endgroup$ Commented Mar 13, 2022 at 22:35