Definition:Reduced Group Word on Set

Definition

Let $S$ be a set.

Let $n \ge 0$ be a natural number.

Let $w = w_1 \cdots w_i \cdots w_n$ be a group word on $S$ of length $n$.

Then $w$ is reduced if and only if:

$\forall i \in \set {1, 2, \ldots, n - 1}: w_i \ne {w_{i + 1} }^{-1}$

Examples

Empty Group Word is Reduced

Let $S$ be a set

Let $\epsilon$ be the empty group word on $S$.

Then $\epsilon$ is reduced.

Also see

• Definition:Reduced Form of Group Word
• Definition:Composition of Reduced Group Words
• Definition:Free Group on Set

• Results about group words can be found here.