Subset/Permutation/Extension/Group homomorphism/Exercise

Let ${}M$ be a finite set, and ${}T\subseteq M$ a subset. Let ${}\operatorname {Perm} \,(T)$ and ${}\operatorname {Perm} \,(M)$ be the corresponding permutation groups. Show that

$\Psi \colon \operatorname {Perm} \,(T)\longrightarrow \operatorname {Perm} \,(M),\varphi \longmapsto {\tilde {\varphi }},$

defined by

${}{\tilde {\varphi }}(x)={\begin{cases}\varphi (x),\,{\text{if }}x\in T,\\x{\text{ else}},\end{cases}}\,$

is an injective group homomorphism.

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