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 $i \in \set {1, \ldots, n - 1}$, which is vacuously true for $\epsilon$.
$\blacksquare$