D4 - cayley's theorem

cayley's theorem

let $G$ be any group, so a group $\bar{G}$ of permutations isomorphic to $G$ needs to be found
for any $g \in G$ define a function $T_{g} : G \to G$ by $T_{g (X)} = g x \forall x \in G$
$T_{g}$ is a permutation on the set of elements of $G$
let $\bar{G} = {T_{g} ∣ g \in G}$ , so it is a group under the operation of function composition
operation preserved: $T_{g} T_{h} (x) = T_{g} (T_{h} (x)) = T_{g} (h x) = g h x = T_{g h} (x)$
$T_{e}$ is the identity, and $(T_{g})^{- 1} = T_{g^{- 1}}$
thus, $\bar{G}$ is a group
for every $g \in G$ , define $ϕ (g) = T_{g}$
if $T_{g} = T_{h}$ , then $T_{g} (e) = T_{h} (e)$ or, $g e = h e$ , thus $g = h$ , ie $ϕ$ is one-to-one
by the way $\bar{G}$ was constructed, it is an onto
let $a, b \in G : ϕ (a b) = T_{a b} = T_{a} T_{b} = ϕ (a) ϕ (b)$ , so it is operation=preserving
therefore, $G ≅ \bar{G}$