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Definition (The Subgroup Generated By A Subset)

Let $G$ be a group. Suppose that $S$ is a subset of $G$.

  1. A word from the alphabet $S$ is a product of the form $$x=s_1s_2\cdots s_k$$ where for each $i$ either $s_i\in S$ or $s_i^{-1}\in S$.
  2. The subgroup generated by $S$ is the set of words from the alphabet $S$, namely $$\left<S\right> = \{s_1s_2\cdots s_k\mid k\in \mathbb{N} \text{ and either $s_i\in S$ or $s_i^{-1}\in S$ for each } i\in\{1,2,\ldots, k\}\}.$$
  3. If $H=\left<S\right>$, then we say that $H$ is the subgroup generated by $S$, or that $S$ is a generating set for $H$.
  4. If $S=\{a\}$, then we'll write $\left<a\right>$ instead of $\left<\{a\}\right>$. We call $\left<a\right>$ the subgroup generated by $a$.

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