What is the meaning of $\langle x\rangle$ in group theory?

Question

Specifically, $\langle x\rangle$ in the text below:

$^*1.$ Let $G$ be a group of order $p^2$, $p$ prime. Prove that either $G$ is cyclic or else it is isomorphic to $\mathbb Z/p\mathbb Z\times\mathbb Z/p\mathbb Z$. [$\textit{Hint:}$ Suppose $G$ contains no element of order $p^2$. Let $x\in Z(G)$, $x\neq1$, and let $y\notin\langle x\rangle$, $y\neq1$. Show that $G\cong\langle x\rangle\times\langle y\rangle$.]

Answer 1

$\langle x\rangle$ is the subgroup generated by $x$, that is the unique minimal subgroup of $G$ containing $x$. This is the definition.

It is not hard to show that $\langle x\rangle = \{x^i : i\in \mathbb{Z}\}$ (with the convention that $x^0$ is the identity element), which is frequently a more useful characterization.

Answer 2

Here $\langle X\rangle$ is the subgroup generated by $X$, where $X\subseteq G$ is a set. It is the minimal subgroup of $G$ that contains $X$. If $X=\{ x_1, \dots, x_n\}$ is finite, then we write it as $\langle x_1, \dots, x_n\rangle$.