Commutative diagrams for groups

2011-02-16T16:48:29Z

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?

Answer by S. Carnahan for Commutative diagrams for groups

2011-02-17T06:13:11Z

These pictures are a way of writing a group as an algebra over an operad. The little square is the 2-ary operation, and the arrows are an indicator that makes the inputs distinguishable.

John Baez has written about operads using pictures similar to yours. See for example TWF week 191.

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 $x(yz) = (xy)z$.

Commutative diagrams for groups - MathOverflow

Commutative diagrams for groups

Answer by S. Carnahan for Commutative diagrams for groups