7.1. Symmetric Group \(S_3\)

Definition

If \(\sigma: S \mapsto T\) and \(\tau: T \mapsto U\), then the composition of \(\sigma\) and \(\tau\) (also called their product) is the mapping \(\sigma \circ \tau: S \mapsto U\) defined by the means of \(s(\sigma \circ \tau) = (s\sigma) \tau \quad \forall s \in S\).

• Let \(G = S_3\), the group of all 1-1 mappings of the set \(\{x_1, x_2, x_3\}\) onto itself, under the product as specified above. G is a group of oder 6.

• Any mapping \(\sigma\) can be specified as:

\[\begin{split}\sigma : \begin{array}{l} x_1 \rightarrow x_i \\ x_2 \rightarrow x_j \\ x_3 \rightarrow x_k \end{array} \quad i,j,k \in \{1,2,3\}, i \neq j, j \neq k, k \neq i\end{split}\]

• Simpler notation \(\sigma : (x_1, x_2, x_3) \mapsto (x_i, x_j, x_k)\)

• Even simpler notation \(\sigma : (1, 2, 3) \mapsto (i, j, k)\)

• Simplest notation \(\sigma = (i,j,k)\)

• \(S_3 = \{ (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)\}\)

• Define \(e = (1,2,3)\), identity mapping

• Define \(\phi = (2,1,3)\)

• Define \(\psi = (2,3,1)\)

7.1.1. Derivations

• \(\phi^2 = \phi \cdot \phi = (2,1,3) \circ (2,1,3) = (1,2,3) = e\)

• Thus \(\phi^{-1} = \phi\)

• \(\psi^2 = \psi \cdot \psi = (2,3,1) \circ (2,3,1) = (3,1,2)\)

• \(\psi^3 = \psi^2 \cdot \psi = (3,1,2) \circ (2,3,1) = (1,2,3) = e\)

• Thus \(\psi^{-1} = \psi^2\)

• \(\phi\psi = \phi \cdot \psi = (2,1,3) \circ (2,3,1) = (3,2,1)\)

• \(\psi\phi = \psi \cdot \phi = (2,3,1) \circ (2,1,3) = (1,3,2)\)

• Thus \(S_3 = \{ e, \phi, \psi, \phi\psi, \psi\phi, \psi^2 \}\)

Elements of \(S_3\)

Element	Mapping	Inverse	Order
\(e\)	(1,2,3)	\(e\)	1
\(\phi\)	(2,1,3)	\(\phi\)	2
\(\psi\)	(2,3,1)	\(\psi^2\)	3
\(\phi\psi\)	(3,2,1)	\(\phi\psi\)	2
\(\psi\phi\)	(1,3,2)	\(\psi\phi\)	2
\(\psi^2\)	(3,1,2)	\(\psi\)	3