[GrayNess]

This is not for a course, simply for my own personal knowledge. I've been dabbling with the Banach-Tarski paradox, which states that if you have a solid 3-D ball, you can break it apart to make two balls identical as the first. At first this may seem to be nonsense and counter-intuitive but mathematically, it works.

As a warning, if you're very bad at math, then looking further on will probably leave you confused and possibly with a headache.

The proof for this paradox is interesting and I understand most of it but am confused on one part. It uses abstract algebra and generators, which takes a bit of time to wrap your head around. G represents a group and S represents a subset of G. So the idea is if S is a subgroup of G, then <S> is a subgroup generated by S, which is the smallest subgroup of G. So G is a subset of S and so, it is a result of however many S there are and S^-1 there are. After a while, you can make some sense of this.

Now for the paradox, it uses generators but here is where I'm confused. It introduces a and b (and their inverses) as generators. For some reason, in a string of a, a^-1, b and b^-1, you cannot have a next to a^-1 and cannot have b next to b^-1. So my question is, why can these not be next to each other?

The rest of the proof I can wrap my head around somewhat except for this part.

[Perna]

They cancel out. They're "allowed" they just cancel out.

http://www.leuschke.org/uploads/Banach-Tarski.pdf