Why is Multiplicative Notation Used for Groups (Instead of Additive)?

Question

In documents relating to group theory it seems common to use a multiplicative notation to represent the group operation. For example, I'm reading Herstein's "Topics in Algebra" and looking for some pointers about vector spaces in the section on groups. In the vector space section the group operation combining vectors is (quite logically) represented as addition ($v = v_1 + v_2$), but when I switch across to the section on groups it's multiplication ($c = ab$).

Besides finding the switch of notation unhelpful, I feel that the additive notation is a better analogy with real arithmetic: all group elements have an inverse as they do with addition, whereas the multiplicative notation carries an untrue suggestion that there may be a "0" which has no inverse.

So, have I missed something: is there some reason why the multiplicative notation is preferable ?

Answer 1

Generally, multiplicative notation is used for the operation in an arbitrary group, and additive notation is reserved for the operation in Abelian (commutative groups). For some intuition into this, the general linear group $GL_n(\mathbb R)$ is the set of all invertible $n\times n$ matrices, with the operation of matrix multiplication. If $A,B\in GL_n(\mathbb R)$ then we would denote their product $AB$ or $BA$ (note that these are not equal in general!). If we consider $\mathbb R^n$ as a vector space, then it is an Abelian group under addition - so if $x,y\in\mathbb R^n$ then we write the sum as $x+y$ or $y+x$ (and these are indeed equal).

Answer 2

You surely know that many groups are not commutative. Because of this, an additive notation would bring confusion in non-abelian context.

Think about group of invertible matrices $\mathrm{GL}(\mathbf{K},n)$ over a field $\mathbf{K}$: this is a group under matrix multiplication: wouldn't be rather tricky to think $A+B$ as the result of $A\cdot B$, knowing that it can be very different from $B+A$?

Inverse elements are also another fact. In general, you can prove that for a group $G$ if $x,y\in G$ then $(xy)^{-1} = y^{-1} x^{-1}$. Imagine this in additive notation: $$ -(x+y)=-y - x$$ and that would be DIFFERENT from $$-x-y=-(y+x)$$

Answer 3

Let $(G,\star)$ be a group with $x\in G$. Suppose you want to write $x \star x \star \cdots \star x$ ($n$ times).

How would you write it with the multiplicative notation? $x^n$.

How would you write it with the additive notation? You can't say $nx$ or $xn$. You have to use an awkward construction like $\sum_{i = 1}^n x$.

Now lets assume we are working over a vector space with scalars in $\mathbb{R}$. How would we write it now? We would just use $nx$. In contexts where there is a reasonable definition to $nx$, usually the additive notation is used.

Add this to the fact that $+$ is used to mean an abelian groups.