# Definition:Word (Abstract Algebra)

This page is about the product of a finite number of elements of a given subset of an algebraic structure. For other uses, see Definition:Word.

## Definition

Let $\struct {G, \circ}$ be a magma.

Let $S \subseteq G$ be a subset.

A word in $S$ is the product of a finite number of elements of $S$.

The set of words in $S$ is denoted $\map W S$:

$\map W S := \set {s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in S, 1 \le i \le n}$

Note that there is nothing in this definition preventing any of the elements of $S$ being repeated, neither is anything said about the order of these elements.

## Also denoted as

Some sources use $\operatorname {gp} S$ for $\map W S$.

## Context

It is usual for the algebraic structure in question to be a group or sometimes semigroup.

If the operation $\circ$ is not associative then this definition still holds.

## Examples

### Set with $2$ Elements

Let $G$ be a group.

Let $X \subseteq G$ be a subset of $G$ such that $X = \set {a, b}$.

Then some of the elements of the set of words $\map W S$ of $G$ are:

$a, b, a b, b a, a b a, b a b, a^{-1} b, b a^{-1}, a b^{-1}, b^{-1} a, a b^{-1}, a^{-1} b^{-1}, a^{-1} b^{-1} a, \ldots$

### Symmetric Group on $3$ Letters

Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:

$\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$

Consider the element $\tuple {132}$.

Then some of the elements of the set of words of $S_3$ which give $\tuple {123}$ are:

$\tuple {123}^2, \tuple {123}^{-1} \tuple {1 2}^2, \tuple {123} \tuple {1 2} \tuple {123}^{-1} \tuple {1 2}^{-1}, \ldots$