What does $\langle X\rangle$ mean? My notes has the following.
Let $G$ be a group. Let $J = \{G_i\}_{i∈I}$ to be the collection of all subgroups of $G$ containing the subset $X ⊆ G$. We note that $G$ is nonempty, since $G$ itself contains $X$, and so $G$ is in $G$. This allows us to set $\langle X\rangle$ as follows: $$\langle X \rangle = \bigcap_{i∈I}G_i$$ 
My understanding is that $\langle X\rangle$ is all the subgroups of $G$ such that those subgroups all contain the elements of $X$. Is that true? Someone please explain this.
 A: $\langle X \rangle$ is a subgroup, not a set of subgroups; in particular, it is the largest subgroup that is contained in every $G_i$. It is also called the subgroup generated by the set $X$, and is the smallest subgroup of $G$ that contains every element of $X$, as well as their products and inverses.
A: $\langle X\rangle$ is the smallest subgroup of $G$ which contains $X$.  It is usually referred to as the subgroup generated by $X$.  It can also be defined as
$$\langle X\rangle=\{x_1^{a_1}\ldots x_n^{a_n}:x_k\in X,a_k\in\Bbb Z, 1\le k\le n\}. $$
A: This is the common notation and it means the subgroup generated by $X$. While the intersection of subgroups is a subgroup as well this is indeed the subgroup of $G$. Your definition is via "outer" approach: one can also define $<X>$ from "the inside" it consists of all expressions $\{x_1^{\varepsilon_1}...x_k^{\varepsilon_k}: x_j \in X, \varepsilon_k \in \{-1,1\}, k \in \mathbb{N}\}$.
A: One thing you seem to be misreading: each $G_i$ is a subset of $G$, and the intersection of a collection of subsets is also a subset. 
Furthermore, each $G_i$ is a subgroup of $G$, and the intersection of a collection of subgroups is also a subgroup, so the right hand side $\cap_{i \in I} G_i$ is a subgroup. So, by definition, $\langle X\rangle$ is a subgroup of $G$.
Furthermore, since each $G_i$ contains $X$, it follows that the subgroup $\cap_{i \in I}G_i$ contains $X$. So it is the smallest possible subgroup containing $X$, because it is contained in every other subgroup $G_i$ that contains $X$.
