What does $\langle X\rangle$ mean when $\langle X\rangle$ is set?

Question

I know that if $a$ is a number, then $\langle a\rangle = \{a^k | k\in Z\}$ (the brackets denotes generator). But what does it mean when that element is replaced by a set? Does it mean the set of all sets $\langle a\rangle$ where $a \in X$?

Answer 1

If $X$ is a subset of a group $G$, then usually $\langle X \rangle$ denotes the subgroup of $G$ generated by $X$ or equivalently the smallest subgroup of $G$ containing $X$ (i.e. $\langle X \rangle= \cap \{S\,|\,S\leq G,X \subseteq S\}$). If one prefers one can build it inductively $X_0=\{e\}\cup X$, $X_n = \{x^{-1}y\,|\,x,y \in X_{n-1}\} \cup X_{n-1}$, then $\langle X \rangle = \cup \{X_i|i\in \mathbb{N}\}$.