Power of Product of Commutative Elements in Semigroup/Examples

Examples of Use of Power of Product of Commutative Elements in Semigroup

Elements of $3$rd Symmetric Group

Let $S = \set {1, 2, 3}$.

Let $S_3$ denote the symmetric group on $3$ letters.

Let $\rho, \sigma \in S_3$ defined in two-row notation as:

\(\ds \rho\)

\(=\)

\(\ds \dbinom {1 \ 2 \ 3} {2 \ 3 \ 1}\)

\(\ds \sigma\)

\(=\)

\(\ds \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2}\)

Then:

$\rho^2 \sigma^2 \ne \paren {\rho \sigma}^2$