is it the operation symbol? is it the equation that defines that symbol? (such as $a\star b=a+b+ab$) or what is it if it's none of the above
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$\begingroup$If $G$ denotes the set on which the group law is defined, then the group law is the map $f : G \times G \rightarrow G$ such that for every $x,y \in G$, $f(x,y) = x*y$.
$\endgroup$ $\begingroup$This might help. It could also make things worse. The example you give uses the real numbers, but transports the structure.
Given a group, we can take any bijection $\varphi$ to (or from...) some slightly different base set, and create a group.
Namely, the real numbers not equal to zero are a group under multiplication.
In the direction I thought of first, our first set will be $\mathbb R \setminus \{-1 \} ,$ while the second set is $\mathbb R \setminus \{0 \} .$ The second set is already a group. We pull back the group operation using, for any $x \neq -1,$$$ \varphi(x) = x+1 \; , \; $$$$ x \star y = \varphi^{-1} \left( \varphi(x) \cdot \varphi(y) \right) $$I deliberately typed in a $\cdot$ to show where the original group operation happens, that being real multiplication. We get$$ x \star y = (x+1)(y+1) - 1 = xy + x + y $$
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