# Let H={sigma in S_5 | sigma94)=4} Show that H<=S_5

Let $H=\left\{\sigma \in {S}_{5}\mid \sigma 94\right)=4\right\}$
Show that $H\le {S}_{5}$
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Roosevelt Houghton

$\sigma$ is bijective map from $\left\{1,2,3,4,5\right\}\to \left\{1,2,3,4,5\right\}$
but $\sigma \left(4\right)=4$
$\sigma$ is bijective map from $\left\{1,2,3,4,5\right\}\to \left\{1,2,3,4,5\right\}$
If $f,g\in H$
$f{g}^{-1}\left(4\right)=f\left({g}^{-1}\left(4\right)\right)=f\left(4\right)=4$
$f{g}^{-1}\left(4\right)=4$
$f{g}^{-1}\in H$