Find the order of the following elements in ${\rm Inn}(S_5)$

Question

Find the order of the following element in ${\rm Inn}(S_5)$: $\phi_{(1243)}.$

So the elements in ${\rm Inn}(S_5)$ are functions $\phi_{(1243)}: S_5 \to S_5$ via $x \to (1243)x(1243)^{-1}$. So the order of this function is the smallest integer $n$ such that $\phi^n = \epsilon$ (the identity map).

I'm confused and don't really understand what I need to do here.

Answer 1

For $x\in S_5$, $(\phi_x)^n=\phi_{x^n}$. So if $(\phi_x)^n=\iota$, then $\phi_{x^n}(g)=g$ for all $g\in S_5$, that is, $x^ng(x^n)^{-1}=g$ for all $g\in S_5$. Therefore $x^n\in Z(S_5)$. Since $Z(S_5)=\{1\}$, we conclude that $x^n=1$. Hence the order of $\phi_x$ in $\operatorname{Inn}(S_5)$ is the same as the order of $x$ in $S_5$.

For your question, $|\phi_{(1243)}|=|(1243)|=4$.