Exploring Non-Associative Groups: Examples, Applications, and Recommended Reads

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In summary, the conversation discusses the existence of groups, loops, and quasigroups and their definitions and applications. The nonzero octonions are given as an example of the best way to define multiplication in an 8-dimensional vector space. Other examples and applications mentioned include string theory.
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Winzer
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Do they exist? What are some examples? Are there any applications?
What are some good books on the topic?
 
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  • #2
Google gives a few articles though they appear fairly advanced.
 
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  • #3
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.)
 
  • #4
Thank you for your responses.
This is something I have been thinking about for a long time-- and have gotten nowhere with.
 

FAQ: Exploring Non-Associative Groups: Examples, Applications, and Recommended Reads

What are non-associative groups?

Non-associative groups, also known as non-associative algebraic structures, are mathematical objects that do not follow the associative property. This means that the order in which operations are performed affects the outcome.

How are non-associative groups different from associative groups?

In associative groups, the order in which operations are performed does not affect the outcome. However, in non-associative groups, the order of operations does matter and can result in different outcomes.

What are some examples of non-associative groups?

Some examples of non-associative groups include quasigroups, loops, and non-associative rings. Non-associative groups can also be found in physics, such as the octonions used in the study of particle physics.

What are some applications of non-associative groups?

Non-associative groups have various applications in mathematics, physics, and computer science. They are used in the study of symmetry, coding theory, and the design of efficient algorithms. They also have applications in fields such as quantum mechanics and relativity.

What are the challenges in studying non-associative groups?

One of the main challenges in studying non-associative groups is the complexity of their structures and operations. This can make it difficult to find general properties and make predictions about their behavior. Additionally, the lack of a simple and consistent definition for non-associative groups can also pose challenges for researchers.

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