# Definition:Reduced Form of Group Word

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## 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:

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