The Subgroup Generated By S Equals The Span Of S

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Problem 66 (The Subgroup Generated By S Equals The Span Of S)

Let $G$ be a group. Suppose that $S$ is a subset of $G$. To parallel our definition of the span of a set of permutations, we could have defined the span of $S$ to be $$\text{span}(S)= \{t_1^{n_1}t_2^{n_2}\cdots t_j^{n_j}\mid j\in \mathbb{N} \text{ and $t_i\in S$ with $n_i\in \mathbb{Z}$ for each } i\in\{1,2,\ldots, j\}\}.$$ Instead we defined the subgroup generated by $\langle S \rangle$ to be $$\left<S\right> = \{s_1s_2\cdots s_k\mid k\in \mathbb{N} \text{ and either $s_i\in S$ or $s_i^{-1}\in S$ for each } i\in\{1,2,\ldots, k\}\}.$$ Using the definitions above, prove that these two sets are the same, so prove $\text{span}(S)=\left<S\right>$. In other words, prove that the subgroup generated by $S$ and the span of $S$ are precisely one and the same.

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