Inner automorphisms of an alternating group

#group_theory

Let $A_{5}$ denote the alternating group on a $5$ -element set ${1, 2, 3, 4, 5}$ . The set of automorphisms of $A_{5}$ form a group, denoted $Aut (A_{5})$ . The group of conjugations of $A_{5}$ , denoted $Conj (A_{5})$ , is the subgroup of $Aut (A_{5})$ consisting of automorphisms of the form $γ_{s} := s (-) s^{- 1}$ where $s \in A_{5}$ . Explicitly, $γ_{s} (x) = s x s^{- 1}$ for any $x \in A_{5}$ .

Prove that the function $γ : A_{5} \to Conj (A_{5})$ , taking $s \in A_{5}$ to $γ_{s}$ , is a surjective homomorphism.
Prove that $A_{5}$ is isomorphic to $Conj (A_{5})$ .

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Let $A_5$ denote the alternating group on a $5$-element set $\{1,2,3,4,5\}$. The set of automorphisms of $A_5$ form a group, denoted $\operatorname{Aut}(A_5)$. The group of {\bfseries conjugations} of $A_5$, denoted $\operatorname{Conj}(A_5)$, is the subgroup of $\operatorname{Aut}(A_5)$ consisting of automorphisms of the form $\gamma_s:=s(-)s^{-1}$ where $s\in A_5$. Explicitly, $\gamma_s(x)=sxs^{-1}$ for any $x\in A_5$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the function $\gamma:A_5\to \operatorname{Conj}(A_5)$, taking $s\in A_5$ to $\gamma_s$, is a surjective homomorphism.
	\item Prove that $A_5$ is isomorphic to $\operatorname{Conj}(A_5)$.
\end{enumerate}