Power of Product of Commutative Elements in Semigroup/Examples/Elements of 3rd Symmetric Group

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

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

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:

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

$\ds \rho^2 \sigma^2$

$=$

$\ds \dbinom {1 \ 2 \ 3} {2 \ 3 \ 1} \dbinom {1 \ 2 \ 3} {2 \ 3 \ 1} \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2} \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2}$

$\ds $

$=$

$\ds \dbinom {1 \ 2 \ 3} {3 \ 1 \ 2} \dbinom {1 \ 2 \ 3} {1 \ 2 \ 3}$

$\ds $

$=$

$\ds \dbinom {1 \ 2 \ 3} {3 \ 1 \ 2}$

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

$=$

$\ds \paren {\dbinom {1 \ 2 \ 3} {2 \ 3 \ 1} \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2} }^2$

$\ds $

$=$

$\ds \dbinom {1 \ 2 \ 3} {3 \ 2 \ 1}^2$

$\ds $

$=$

$\ds \dbinom {1 \ 2 \ 3} {1 \ 2 \ 3}$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $9$