Consider the following statements:
$\text{(A)}$ Let $G$ be a group and let $H \subset G$ be a subgroup of index 2 . Then $[G, G] \subseteq H$.
$\text{(B)}$ Let $G$ be a group and let $H \subset G$ be a subgroup that contains the commutator subgroup $[G, G]$ of $G$. Then $H$ is a normal subgroup of $G$.
Which of the following statements is correct?
- $\text{(A)}$ and $\text{(B)}$ are both true
- $\text{(A)}$ and $\text{(B)}$ are both false
- $\text{(A)}$ is true and $\text{(B)}$ is false
- $\text{(A)}$ is false and $\text{(B)}$ is true
