TITLE: Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?
QUESTION [5 upvotes]: Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial abelian group can satisfy this property, so I suppose the first question is: Do such groups $G$ exist?

REPLY [10 votes]: Theorem The only finite group satisfying the condition is the trivial group.
Proof. Let $G$ be a nontrivial finite group and $S$ be a simple quotient of $G$, which satisfies the condition by Gaschutz's lemma. Then $S$ is non-abelian as mentioned above. If $x\in S$ is any involution, then by  well-known result of Guralnick and Kantor in Probalistic generation of finite simple groups, there exists an element $y\in S$ such that $S=\langle x,y\rangle$ while $\langle x,yxy^{-1}\rangle$ is a dihedral group, that is, $S\neq\langle x,yxy^{-1}\rangle$, a contradiction.