Word (Abstract Algebra)/Examples/Symmetric Group on 3 Letters

Examples of Words in the Context of Abstract Algebra

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 subset $\set {\tuple {1 2}, \tuple {132} }$ of $S_3$.

Then some of the elements of the set of words of $\set {\tuple {1 2}, \tuple {132} }$ are:

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

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset: Example $96$