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 ?
Best Answer
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 $$-x-y=-(y+x)$$