Understanding of a proof

Question

In (1) and (3), $ M' \subseteq L'$ and $H \subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' \leq L'$ and $H \leq H''$ again respectively.

Answer 1

I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' \subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' \leq L'$.

Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H \subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H \leq H''$.