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 $G\cong\langle x\rangle\times\langle y\rangle$.]
Best Answer
$\langle x\rangle$ 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.