Two elements $a$ and $b$ of a group $G$ are conjugate in $G$ if there is an element $g$ in $G$ such that $g^{-1}·a·g\=b$ . Any such element $g$ is called a conjugator.  A conjugator is not generally uniquely determined by $a$ and $b$.

The AreConjugate( a, b, G ) command returns true if the permutations a and b are conjugate in the permutation group G, and returns false otherwise.

The Conjugator( a, b, G ) command returns an element g in G such that g^(-1) . a . g = b, provided that a and b are conjugate in G. If a and b are not conjugate in G, the value FAIL is returned.

The group G must be an instance of a permutation group, and the permutations a and b must be members of G.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{PermutationGroup}\left(\mathrm{Perm}\left(\left[\left[2\,4\,6\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,5\right]\,\left[2\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,4\right]\,\left[2\,5\right]\,\left[3\,6\right]\right]\right)\right)$

$G≔⟨\left(2\,4\,6\right)\,\left(1\,5\right)\left(2\,4\right)\,\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)⟩$

$a≔\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\,\left[5\,6\right]\right]\right)$

$a≔\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)$

$b≔\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,6\right]\,\left[4\,5\right]\right]\right)$

$b≔\left(1\,2\right)\left(3\,6\right)\left(4\,5\right)$

$\mathrm{AreConjugate}\left(a\,b\,G\right)$

$\mathrm{false}$

$\mathrm{AreConjugate}\left(a\,b\,\mathrm{Symm}\left(6\right)\right)$

$\mathrm{true}$

$c≔\mathrm{Perm}\left(\left[\left[1\,4\right]\,\left[2\,5\right]\,\left[3\,6\right]\right]\right)$

$c≔\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)$

$\mathrm{AreConjugate}\left(a\,c\,G\right)$

$\mathrm{true}$

$d≔\mathrm{Conjugator}\left(a\,c\,G\right)$

$d≔\left(2\,4\,6\right)$

$d^{-1}·a·d=c$

$\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)=\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)$

The GroupTheory[AreConjugate] and GroupTheory[Conjugator] commands were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.