Let $H$ be a subgroup of a group $G$. Prove that $\langle H\rangle =H$.
My solution goes like this:
We know that $\langle H\rangle$ is the intersection of all the subgroups of $G$ containing $H$, or it is the smallest subgroup of $G$ containing $H$. So from this definition, $H\subset \langle H\rangle$. Now $H$ is also a subgroup of $G$ containg $H$. Now if $g_1\in G$ and $g_1\notin H$ then $g_1\notin\text{the intersection of all subgroups containing H}$. Hence, if $g_1\in \langle H\rangle$, then $g_1\in H$. Hence $ \langle H\rangle\subset H$. Thus $H= \langle H\rangle$.
Is the above solution correct? Is it valid? If not, where is it going wrong? There may be many posts on this site concerning the same topic, but I wanted to verify whether my solution is correct or not. I found a particular thread: Prove that if $H$ is a subgroup of $G$ then $\langle H \rangle= H$. But it asks for a different proof verification .
Best Answer
It looks fine. But since you point out that $\langle H\rangle$ is the smallest subgroup containing $H$, we can argue more easily (basically the same):
$H$ is the smallest set containing $H$, and it is a subgroup, so it is the smallest subgroup containing $H$.