Do Non-Associative Groups Exist and What Are Their Applications?

[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 on Wikipedia has a nice little table of what you call groups minus this or that axiom.

The nonzero octonions 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 (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.