# Definition:Reduced Form of Group Word

## Definition

Let $X$ be a set.

Let $w$ be a group word on $X$.

The reduced form $\map {\operatorname {red} } w$ of $w$ is the unique reduced word for which there exists a reduction:

$w = w^{\paren 0} \to w^{\paren 1} \to \cdots \to w^{\paren n} = \map {\operatorname {red} } w$