\begin{definition}[Definition:Elementary Reduction of Group Word on Set]
Let $X$ be a set.
Let $v$ and $w$ be group words on $X$.
Let $n$ be the length of $v$.
Then $w$ is an '''elementary reduction''' of $v$ {{iff}}:
:$w$ has length $n - 2$
:There exists $k \in \set {1, \ldots, n - 1}$ such that:
::$v_k = v_{k + 1}^{-1}$
::$w_i = \begin {cases} v_i & : i < k \\ v_{i + 2} & : i > k + 1 \end {cases}$
This is denoted $v \overset 1 \longrightarrow w$.
\end{definition}