# Empty Group Word is Reduced

## 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 $\epsilon$.

$\blacksquare$