What kind of algebraic structure is a group with the conjugation as a second operation? It is interesting in a group that conjugation $(x,y)\mapsto x^y:=y^{-1}xy$ behaves like exponentation among real numbers:


*

*$(xy)^z=x^zy^z$

*$(x^y)^z=x^{yz}$


That is, real numbers with multiplication and exponentation forms the same algebraic structure as a group with multiplication and conjugation  Is there a name for this algebraic structure?
 A: The two laws of exponentiation you noted are both studied in group theory not as an operation on a set but rather in the context of group actions.
The second law, $(x^y)^z = x^{yz}$, is saying that a group acts on itself. This is a particular instance of a more general situation, where a group could be acting on a set/vector space/other group. 
The first law, $(xy)^z = x^z y^z$, is saying that this action respects the structure of the set on which the group acts. In your case, the axiom is saying that the action respects multiplication on the group. 
In a similar way, you could have a group action on a vector space by linear transformations, and the conditions for this to hold would be expressible in terms of analogous axioms you noted for a group acting on itself.
A: In addition to darko's answer, a group acting on itself by conjugation is a special case of crossed module.
A crossed module is defined by


*

*two groups $G$ and $H$

*a group homomorphism $\partial : G\to H$

*an group action of $H$ on $G$, denoted $g^h$ for all $g\in G$ and $h\in H$, which respects the group structure of $G$ (as explained in darko's answer)


satisfying two axioms for all $g,g'\in G$ and $h\in H$:


*

*$\partial (g^h) = h^{-1}\partial(g)h$

*$g^{\partial (g')}=g'^{-1}gg'$ (this is called Peiffer's identity).


When $G=H$, $\partial(g)=g$ and $g^h=h^{-1}gh$, these two axioms are trivially satisfied.
