Exercise 3.1.22 (Intersection of a collection of normal subgroups)

Proof. Let $G$ be a group.

(a)

Suppose that $H,K$ are normal subgroups of $G$. Then $H\cap K$ is a subgroup. Let $a\in H\cap K$. For all $g\in G$, since $a\in H$ and $H⊴G$, $\mathit{ga}{g}^{-1}\in H$, and similarly $\mathit{ga}{g}^{-1}\in K$, so $\mathit{ga}{g}^{-1}\in H\cap K$. This shows that $$H\cap K⊴G.$$

(b)

Let ${(H_i)}_{i\in I}$ be a family of normal subgroups of $G$, where $I\ne \varnothing$ is any set. Then $H=\underset{i\in I}{\bigcap <!--\; FUNCTION\; APPLICATION\; -->}{H}_i$ is a subgroup of $G$.

Suppose that $g\in G$ and $a\in \underset{i\in I}{\bigcap <!--\; FUNCTION\; APPLICATION\; -->}{H}_i$. For each $i\in I$, $a\in H_i$, where $H_i⊴G$, thus $\mathit{ga}{g}^{-1}\in H_i$. This is true for every $i\in I$, so

$$\mathit{ga}{g}^{-1}\in \underset{i\in I}{\bigcap }{H}_i.$$

This shows that

$$\underset{i\in I}{\bigcap }{H}_i⊴G.$$