# Why is there a hierarchy of interest between associativity and commutativity

In mathematical structures, there are among other things : groups.

Among their particular properties of the group, the groups have the property of associativity. Within the various groups, there are commutative (abelian) and non commutative (non-abelian) groups.

Why is there a hierarchy of interest between associativity and commutativity in groups ; that is , why do we assume that groups are associative, while commutativity is only an "option" ? (why is associativity "more important" than commutativity ?)

Are there algebra structures which don't assume associativity ?

• "Are there algebra structures which don't assume associativity ?" That's the answer. If you do not know any, probably they are less interesting. Nonetheless, I give you an elementary but important example: Lie algebras. They are not associative, but they are "almost" commutative in some sense.
– A.G
Commented Jun 27, 2020 at 16:51
• Associativity is "more important" than commutativity because of (linear) maps. In fact, the composition of maps is automatically associative but not necessarily commutative. The most prominent example of such an algebra is $M_n(K)$, or on the group level $GL(V)$. Commented Jun 27, 2020 at 16:53

There absolutely are non-associative structures! Relevant terms here include "loop" and "Lie algebra." For concrete examples, consider octonion multiplication or the "midpoint" operation on points in $$\mathbb{R}^n$$ (note that the latter is commutative but not associative!). There are also weakenings of the associative law, like power associativity or alternativity. The study of nonassociative structures is very much a thing.

However, there is definitely an important sense in which associativity is special, namely that it lines up with function composition (note that function composition is not commutative: $$(x+1)^2\not=x^2+1$$ in general). This is important since we generally run into groups not "in a vacuum" but rather as a component of a richer object - namely a group action. A group action is basically a way of representing elements of a group $$G$$ as "structure-preserving" maps on some other structure $$A$$. The point is that this is generally the way groups emerge in the first place: consider for example Galois theory, where we're not interested in $$Gal(K/F)$$ on its own so much as we're interested in the action of $$Gal(K/F)$$ on $$K$$.

And the other direction is important too: given a group $$G$$ we can often gain a better understanding of $$G$$ by thinking about its possible actions. An early example of this is Cayley's theorem, where we look at a simple action of a group on itself, and representation theory is built around the idea that we can learn a lot about a group by looking at the various ways it can act on vector spaces.

The idea of an action - where elements of the acting structure represent functions on the acted-on structure, and the operation of the acting structure represents composition - automatically enforces associativity since function composition is associative. Conversely, if we try to whip up a notion of "action" which does not automatically enforce associativity - that is, a notion of a non-associative magma acting on a structure $$X$$ in such a way that $$[a*(b*c)]x\not=[(a*b)*c]x$$ in general - things get messy. This doesn't make non-associative structures uninteresting, but it does mean that the natural idea of "action" is really only appropriate to associative structures, and this does wind up being a big deal.

"Are there algebra structures which don't assume associativity ?" Yes, of course. See for example Lie algebras, Jordan algebras or more generally non-associative algebras. This includes also the octonions, pre-Lie algebras, post-Lie algebras and several other non-associative algebras.

• Thank you so much Commented Jun 27, 2020 at 17:29

The definition of (abstract) group is historically patterned upon the properties of the bijections on a set: any three of them fulfill the associative property, but not every pair of them commute.