# Commutative diagrams for groups

You can present a group in a Cayley-like manner, replacing colors by explicit assignment of nodes to edges: while in a Cayley graph $x \circ y = z$ is presented like this:

you can also present it like this:

Now the group axioms can be stated like this:

• For each $x, y$ there is a unique $z$ with $x \circ y = z$, or: such that the following diagram holds ("commutes"):

• There is an $e$ such that for all $x$ it holds that $x\circ e = e \circ x = x$, or: such that the following two diagrams commute:

and for each $x$ there is a $x^{-1}$ such that $x \circ x^{-1} = x^{-1} \circ x = e$, or: such that the following diagram commutes:

• For each $x, y, z$ it holds that $x \circ (y \circ z) = (x \circ y) \circ z$, or: such that the following diagram commutes:

The last diagram is somewhat ugly, even when drawn in this most balanced way (I didn't find a more appealing and symmetric one).

But an astonishing symmetry arises, when we consider Abelian groups. Commutativity is expressed by the diagram:

and associativity becomes:

In the presence of commutativity, associativity seems to be related to commutativity (some sort of "second level commutativity").

Can any use be made of this kind of diagrams, or is it just vain baublery?

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I guess your diagram for associativity in general groups is just a kind of ugly, because you draw it in two dimensions. In three dimensions you can put the squares as corners of tetrahedron and the circles on the edges ;-). (OK, the directions of the arrows are not so clear.) – Someone Feb 16 '11 at 17:51
It looks a bit like "sketches" to me, but I'm not an expert so shall merely mention it. As for your comment about commutativity and associativity, associativity is commutativity. It's commutativity of the operations $L_x$ and $R_z$ where $L_x$ is "multiply on the left by $x$" and $R_z$ is (you've guessed it), "multiply on the right by $z$". I have a vague memory of seeing an article going into some detail about this. Under commutativity, $R_z = L_z$ so we can rewrite $x(y z)$ as $x (z y)$ and $(x y) z$ as $z (x y)$ whereupon associativity becomes commutativity of $L_x$ and $L_z$. – Loop Space Feb 16 '11 at 18:23
I vote for baublery. – Martin Brandenburg Feb 17 '11 at 0:12

The relation between associativity and commutativity is similar to a fact seen in some books on vertex algebras, where one starts with an axiom like $x(yz) = y(xz)$ and after some power series manipulations deduces $x(yz) = (xy)z$.