I have a doubt ${\color{red}{\text{(the red part below)}}}$ in Dummit&Foote's Abstract Algebra on page134 :
Corollary 14. If $K$ is any subgroup of the group $G$ and $g\in G$ , then ${\color{red}{K\cong gKg^{-1}}}$ . Conjugate elements and conjugate subgroups have the same order.
Proof: ${\color{red}{\text{Letting }G=H}}$ ${\color{red}{\text{ in the proposition shows that}}}$ ${\color{red}{\text{conjugation by }g\in G}}$ ${\color{red}{\text{ is an automorphism of }G\ ,}}$ from which the corollary follows.
And the proposition in the above paragraph is:
Proposition 13. Let $H$ be a normal subgroup of the group $G$ . Then $G$ acts by conjugation on $H$ as automorphisms of $H$ . More specifically, the action of $G$ on H by conjugation is defined for each $g\in G$ by $$ h\mapsto ghg^{-1}\ \ \text{for each }h\in H\ . $$ For each $g\in G$ , conjugation by $g$ is an automorphism of $H$ . The permutation representation afforded by this action is a homomorphism of $G$ into with kernel $C_G(H)$ . In particular, $G/C_G(H)$ is isomorphic to a subgroup of $\mathrm{Aut}(H)$ .
What I can know is just that letting $G=H$ in proposition13 we obtain $G=gGg^{-1}$ , but how can it ${\color{red}{\text{(the sentence in red)}}}$ conclude that $K\cong gKg^{-1}$ ? I could not find the connection with them although it is easy to check that $K\cong gKg^{-1}$ is true in other ways.
Proposition 13, applied with $G = H$, tells you that the function $$\phi : G \to G, \ \ \ \ \ \phi(x) = g x g^{-1}$$ is an isomorphism, for any $g \in G$.
Restricting $\phi$ to the subgroup $K \subset G$, we obtain an isomorphism $$ \phi|_K : K \to \phi(K), \ \ \ \ \ \phi(x) = g x g^{-1}. $$
The image $\phi(K)$ is precisely $gKg^{-1}$. Thus $K$ is isomorphic to $gKg^{-1}$, via the isomorphism $x \mapsto gxg^{-1}$.