[tzs]

You are right that it is not really an algebra thing. It's a bijection between finite sets thing. If you apply a bijective mapping from set A onto set B, then apply some permutation to B, and then apply the inverse mapping, you get a permutation of A that has the same structure as the permutation you applied to B.

By "same structure" I mean that if written as a product of disjoint cycles has the same number and sizes of cycles.

Abstract algebra is probably the first place most people would encounter it though, in the context of conjugates in groups.

[TomSwirly]

> it is not really an algebra thing. It's a bijection between finite sets thing.

Mathematician here. Even just a bijection between finite sets is in fact algebra.

But the space of all Rubik's Cube positions is in fact a finite non-commutative group - right smack-dab in the middle of the field of algebra.