TITLE: characterization of the operation on a finite or infinite group.
QUESTION [5 upvotes]: Suppose $G$ is a group which is a group again with another operation $*$. For each $a,b\in G$ with $a\ne b$ we have:
$$ab=a*b$$
Can always the condition $a\ne b$ be dropped?

REPLY [7 votes]: Yes. This follows from the fact that for any $a\in G$ and any group operation multiplication by $a$ is a bijection. Thus we have two bijections
$$
\cdot_a:G\rightarrow G\text{ and }\star_a:G\rightarrow G,
$$
the first mapping $g\in G$ to $a\cdot g$ and the second to $a\star g$.
The two bijections are the same on $G\setminus\{a\}$. Since they are bijections, the image of $G\setminus\{a\}$ must be $G\setminus\{b\}$ for some $b\in G$ and 
$$
a\cdot a=b=a\star a.
$$