Non-Associative groups

[Winzer]

Do they exist? What are some examples? Are there any applications?
What are some good books on the topic?

[NoMoreExams]

Google gives a few articles though they appear fairly advanced.

[Marcaias]

A group's operation is associative by definition. If you take out the associativity axiom, you get what's (apparently) called a loop. If you also remove the need for an identity, you get a quasigroup. This page (http://en.wikipedia.org/wiki/Semigroup) on Wikipedia has a nice little table of what you call groups minus this or that axiom.

The nonzero octonions (http://en.wikipedia.org/wiki/Octonion) under multiplication form a loop. They are an example of the best you can do if you want to reasonably define multiplication on an 8-dimensional vector space over the reals. (The complex numbers are the best you can do in 2-dimensions. There you get everything you could want out of multiplication: it commutes, it associates, it has an inverse. In 4-dimensions you can form the quaternions, but you lose commutativity. In 8-dimensions you also have to lose associativity, and what you get are called octonions.)

As far as applications go, apparently they may be quite important in string theory (http://math.ucr.edu/home/baez/octonions/node1.html) (as far as you can call string theory applicable to anything.)

[Winzer]

Thank you for your responses.
This is something I have been thinking about for a long time-- and have gotten nowhere with.