I have this in my lecture notes, I think it relates to mapping but I'm not sure:
$\bullet{}: G\times G\rightarrow G$
What does that actually mean? I know that for a group the binary operation has to map $G$ to $G$ so $\bullet : G \bullet G \rightarrow G$ would make sense to me, is that what it's meant to mean? And multiplication is being used as an example? Or does the $\times$ mean something different in this context?
I have a definition of a map that says $\Phi : A \rightarrow B $ is a method for associating elements of A to elements of B, which looks somewhat similar to the above. From the perspective of a physicist with no background in group theory, anyway.
Best Answer
If $A$ and $B$ are sets, $A\times B$ denotes the couples of elements in $A$ and $B$. You could write $$A\times B=\left\{(a,b)\mid a\in A, b\in B\right\}.$$ (in principle this is not a definition, but let us not dwell on details)
A map is another name for a function in many situations. So a map $\cdot: G\times G\rightarrow G$ is simply a function that maps a couple of elements in $G$ to an element of $G$.
So indeed, the multiplication on $\mathbb{R}$ does exactly that, given two real numbers it yields another real number.