Is the difference between additive groups and multiplicative groups just a matter of notation?

Question

Groups are sometimes written additively, like $(G,+)$, and sometimes they are written multiplicatively, like $(G,\times)$. But is there really a difference? Is it all just a matter of notation, nothing more? If it is something beyond notation, what is the difference, exactly?

Answer 1

It's just notation. Typically, though, groups written additively are assumed to be abelian.

But any group could just as well be $(G,\star)$ for an appropriate binary operation $\star$.

Answer 2

Shaun's answer is absolutely correct: it's just a matter of notation and convention. When developing the general theory of groups, it is more common to use multiplicative notation, since additive notation is conventially reserved for commutative groups, and groups are not in general commutative. Multiplicative notation also has the advantage that $a\times b$ can be written simply as $ab$, and this advantage becomes more apparent when you start to consider products of three or more terms.

That being said, it's important to familiarise oneself with both notations, since two things which might appear very different might actually have the same meaning. For example, in a group $G$ written additively, if $n\in\mathbb N$ and $g\in G$, we write $ng$ to mean $\underbrace{g+g+\dots+g}_{\text{$n$ times}}$. If $G$ is instead written multiplicatively, then the same thing would be written as $g^n$. This means that "scalar multiplication" and "exponentiation" by natural numbers actually correspond to the same concept: repeated application of the group operation.

Finally, it's worth noting that the term "multiplicative group" has two distinct meanings. The first meaning is that it is a group which happens to be written multiplicatively, so the adjective "multiplicative" is actually describing the notation used to represent the group, rather than the group itself. The second meaning is used in ring theory: if $(R,+,\times)$ is a ring, then its multiplicative group is the group $(R^\times,\times)$, where $R^\times$ is the set of invertible elements of $R$. The multiplicative group of ring is also known as its group of units or unit group. By contrast, the additive group of a ring is the commutative group $(R,+)$. Outside of ring theory, "additive group" might instead just mean a group written using additive notation.

Answer 3

For study of a group, it doesn’t matter much what symbol you use, but if there is another binary operation over the same set, as in a ring or field, you need a different symbol symbol to represent that operation.