# Definition A *multiplication table* is a graphical representation of the multiplication of the products of all [[Group Element|elements]] of a given [[group]]. # Example Consider the group $P(3)$, the permutation group of a set with 3 elements, which has 3! = 6 [[Group Element|elements]]. They can be written as: $ \begin{align} e = \begin{pmatrix} 1 & 2 & 3\\ 1 & 2 & 3 \end{pmatrix}, \quad a = \begin{pmatrix} 1 & 2 & 3\\ 1 & 3 & 2 \end{pmatrix}, \quad b = \begin{pmatrix} 1 & 2 & 3\\ 3 & 2 & 1 \end{pmatrix}, \\ c = \begin{pmatrix} 1 & 2 & 3\\ 2 & 1 & 3 \end{pmatrix}, \quad d = \begin{pmatrix} 1 & 2 & 3\\ 3 & 1 & 2 \end{pmatrix}, \quad f = \begin{pmatrix} 1 & 2 & 3\\ 2 & 3 & 1 \end{pmatrix}, \\ \end{align} $ The multiplication table has $6^2 = 36$ cells. First, we rewrite these permutations in [[cycle notation|cycle notation]]: $ \begin{align} e &= (1)(2)(3) = (),\\ a &= (1)(23) = (23),\\ b &= (2)(13) = (13),\\ c &= (3)(12) = (12),\\ d &= (132),\\ f &= (123),\\ \end{align} $ Then we multiply them in Mathematica because we're lazy and there's 36 of them. The first 12 are: ![[Pasted image 20210626100356.png]] Finally we get the multiplication table: | | $e$ | $a$ | $b$ | $c$ | $d$ | $f$ | | --- | --- | --- | --- | --- | --- | --- | | $e$ | $e$ | $a$ | $b$ | $c$ | $d$ | $f$ | | $a$ | $a$ | $e$ | $d$ | $f$ | $b$ | $c$ | | $b$ | $b$ | $f$ | $e$ | $d$ | $c$ | $a$ | | $c$ | $c$ | $d$ | $f$ | $e$ | $a$ | $b$ | | $d$ | $d$ | $c$ | $a$ | $b$ | $f$ | $e$ | | $f$ | $f$ | $b$ | $c$ | $a$ | $e$ | $d$ |