# Definition:Elementary Reduction of Group Word on Set

## Definition

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 $v$ if and only if:

• $w$ has length $n-2$
• There exists $k \in \{1, \ldots, n-1\}$ 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 $v \overset 1 \longrightarrow w$.