I am searching for examples of groups which contain their own operation as an element. I am having difficulty showing that this is not possible for groups of size greater than 1, but counterexamples are also elusive.
Taking the set $S=\{f\}$
If we define
$f:(S \times S) \to S$
$f(f,f)=f$
Then we can construct a group $G$ on the set $S$ with operation $f$. The identity of $G$ is $f$, and this group satisfies all required properties.
My question is: Are there groups of size greater than 1 which contain their operation as an element?