I know exactly one party trick based on mathematical group theory, which I have actually used to impress non-mathematicians at a party.

There's a concept called the "center" of a "group", which is the set of operations that commute with every other operation in the group. The center always contains the identity operation of doing nothing. The group of scramblings of a Rubik's cube happens to contain exactly two elements in its center: the identity, and a move called the "superflip" which takes a little bit of effort to memorize how to do, but it's not so hard. Much easier than actually solving a scrambled Rubik's cube. It's like you do a simple move repeated 4x, and then you do that whole 4x set 3x with some rotations in between. Not terribly complicated. Importantly, once you memorize it it's not difficult to do just by feel, since it's a fixed sequence of mechanical motions.

So, the party trick goes like this:

You have a Rubik's cube that is exactly a superflip away from the solved state. You hand it to an unsuspecting party guest and say "go ahead and make one or two turns" (it's important to say something like "one or two" because if they do 3 the trick becomes challenging, and if they do 4 or more it might become impossibly difficult unless you're actually good at solving Rubik's cubes, which I am not). They take this obviously unsolved cube and make a couple more moves so now it appears even more scrambled.

You take the cube back and do the superflip behind your back, without looking at the cube.

Then you move the cube out from behind your back, and at the same time (trying to be slick about it) you undo the one or two moves remaining before it is solved. Everyone gasps and say "omg he solved it behind his back" (when really you did no such thing).

This works because if S is the superflip and X is the simple moves they did to it, S X S is equal to just X because S commutes with everything. (S is also its own inverse, so that S S = 1.)