Empty Group Word is Reduced
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Theorem
Let $S$ be a set
Let $\epsilon$ be the empty group word on $S$.
Then $\epsilon$ is reduced.
Proof
By definition, a group word $w = w_1 \cdots w_i \cdots w_n$ is reduced if and only if:
- $w_i \ne {w_{i + 1} }^{-1}$ for all $i \in \set {1, \ldots, n - 1}$
where $w_1, w_2, \ldots$ are elements of $S$.
We have by hypothesis that $\epsilon$ is the empty group word on $S$.
Hence by definition it has no elements of $S$ in it.
Hence the condition for $\epsilon$ to be reduced is vacuously true.
$\blacksquare$