Computations with inner automorphisms

#group_theory

Let $G$ be a group, and let $Aut (G)$ denote the group of automorphisms of $G$ . There is a homomorphism $γ : G \to Aut (G)$ that takes $s \in G$ to the automorphism $γ_{s}$ defined by $γ_{s} (t) = s t s^{- 1}$ .

Prove rigorously, possibly with induction, that is $γ_{s} (t) = t^{b}$ , then $γ_{s^{n}} (t) = t^{b^{n}}$ .
Suppose $s \in G$ has order 5, and $s t s^{- 1} = t^{2}$ . Find the order of $t$ . Justify your answer.

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Let $G$ be a group, and let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$. There is a homomorphism $\gamma:G\to \operatorname{Aut}(G)$ that takes $s\in G$ to the automorphism $\gamma_s$ defined by $\gamma_s(t)=sts^{-1}$.
	\begin{enumerate}[label=\alph*)]
		\item Prove rigorously, possibly with induction, that is $\gamma_s(t)=t^b$, then $\gamma_{s^n}(t)=t^{b^n}$.
		\item Suppose $s\in G$ has order 5, and $sts^{-1}=t^2$. Find the order of $t$. Justify your answer.
	\end{enumerate}