A group having more than one elements with only one element as inverse of each element in the group.
Is there any name for that? Let me explain my question: $(\{0\}, +)$ is a trivial group with only one element (additive identity). Additive inverse of element $0$ is $0$ in this group. There is only one element and thus an inverse.
Similarly, if we define a group $(G,\wedge)$ with elements $\{0,1,2\}$ and a binary operation $\wedge$ defined as $a \wedge b =a^b \ \forall a,b \in G$. Then it is clear that the identity element in this group is $1$ and inverse of each element is $0$. What should I call this group? I think that there might be a special name for this kind of groups.
edit: Unfortunately I gave a wrong example. But not considering the example here, does there exist such a group with more than one elements where one element works as inverse of all elements?