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 $i \in \set {1, \ldots, n - 1}$

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.

$\blacksquare$