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

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Discussion Overview

The discussion revolves around the existence of non-associative groups, exploring their definitions, examples, and potential applications, as well as resources for further reading. The scope includes theoretical considerations and applications in advanced mathematics and physics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the existence of non-associative groups and seeks examples and applications.
  • Another participant notes that while groups are defined to be associative, removing this axiom leads to structures known as loops and quasigroups.
  • The nonzero octonions are presented as an example of a loop, highlighting their significance in an 8-dimensional vector space and their potential relevance in string theory.
  • Participants express a desire for resources, with one mentioning that available articles appear advanced.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus on the existence and applications of non-associative groups, with multiple viewpoints presented regarding their definitions and implications.

Contextual Notes

Participants mention various mathematical structures that arise from relaxing the associativity axiom, but there are unresolved questions about the implications and applications of these structures.

Winzer
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Do they exist? What are some examples? Are there any applications?
What are some good books on the topic?
 
Physics news on Phys.org
Google gives a few articles though they appear fairly advanced.
 
Last edited:
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.)
 
Thank you for your responses.
This is something I have been thinking about for a long time-- and have gotten nowhere with.
 

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