TITLE: Is the infinite intersection of subgroups still a subgroup?
QUESTION [5 upvotes]: Suppose $H_i < G$ where $G$ is a group, and $H_i$ is a subgroup of $G$. Then it is true that $\bigcap_{i=1}^{\infty} H_i$ still a subgroup?

REPLY [9 votes]: Sure it is. It contains $1$, and then if $a, b \in \bigcap_{i=1}^{\infty} H_i$, then $a, b \in H_i$ for each $i$, so $a b^{-1} \in H_i$ for each $i$, so $a b^{-1} \in \bigcap_{i=1}^{\infty} H_i$.