
> 
> Does there exist a finite group $G$ of order greater than two containing a unique element $g$ such that
> $$
> g\notin\langle x\rangle
> \hbox{ for all $x\in G\setminus\{g\}$ ?} 
> $$
> 
> 
> 


Or we have another [fantastic property of the order-two group](https://mathoverflow.net/q/148925/24165)?


Clearly, such a group must be a 2-group, and the unique lonely element must be central and of order two.



Groups with a unique lonely element
I think that the nontrivial semidirect product of a cyclic group of order 4 $\langle x\rangle$ acting on another cyclic group of order 4 $\langle y\rangle$ is an example of such a group. The center of this group is $\langle x^2,y^2\rangle$ and $x^2y^2$ is not a square.


I came up with this trying to prove that no such group existed. In an example the center cannot be cyclic and then a minimal example had to be like this.


