# Why do we think of group compositions as multiplication?

This has bothered me for some time: The composition in a group is usually denoted $xy$ or $x\cdot y$. Powers (note the word) are denoted by $x^n$, inverses by $x^{-1}$, and the neutral element by $1$.

Someone clearly seemed to think of multiplication when these conventions were adopted.

But wait, in most algebraic (ring-like) structures where multiplication is defined, this operation almost never makes that structure into a group, even if you take away zero. Wouldn't it have been much more natural to use additive notation for groups? Obviously, when we call something "addition," it is usually commutative, but then again, this is mostly a result of conventions; multiplication was also traditionally thought of as commutative until evil people invented non-commutative rings.

Are there any historical/heuristic/practical explanation for this (in my opinion) strange choice of notation?

The best explanation I can come up with is that it works, that non-commutative rings just turned out to be such an interesting topic that people stopped thinking of multiplication as always commutative. Hence they used multiplicative notation when the group was not assumed to be Abelian.

• Well, you're going to need addition and multiplication for a ring anyway... and addition is the commutative operation (whereas multiplication isn't necessarily, still). – Batman Apr 10 '15 at 18:42
• @Batman part of the question seems to be "when did multiplication first become non-commutative"? – Omnomnomnom Apr 10 '15 at 18:43
• You might be able to blame Arthur Cayley; he invented both matrix-multiplication and groups as we know them today. – Omnomnomnom Apr 10 '15 at 18:46
• There's also Hamitlon with quaternion-multiplication – Omnomnomnom Apr 10 '15 at 18:48
• Keep in mind that a lot of the pioneering work in group theory was done in permutation groups, where the operation is composition of permutations (and keep in mind Cayley's theorem) – Bill Dubuque Apr 10 '15 at 19:02

## 2 Answers

I guess the main reason is representation theory: You can represent every group by linear operators on a vector space, where the group operation maps to the multiplication of operators. You cannot generally map the group operation to addition of operators.

A group $G$ is a set endowed with a binary operation, the map $\cdot :G\times G \to G$ that obeys some properties. The notation is then $\cdot(g,h) := g\cdot h$.

Of course, nothing stops us from using the plus symbol for the map $+ : G\times G \to G$. But mathematicians are lazy, and a dot is much easier to write than a plus sign!

Also from a categorical point of view, a group can be thought of as a category with a single object, whose morphisms are all isomorphisms, i.e. a groupoid.

Then if $*$ is the lone object of the groupoid $\mathcal C$, a group is $\text{Aut}_\mathcal{C}(*).$ Its elements are morphisms $f: * \to *$, whose binary operation is composition. Composition is traditionally denoted by $\circ$ or sometimes even $gf$, which is reminiscent of multiplication.

• Well I believe mathematician keep the + sign for commutative groups. – Thibaut Dumont Aug 12 '15 at 20:33