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GroupTheory

 AreConjugate
 test conjugacy of elements of a permutation group
 Conjugator
 compute the conjugator of elements of a permutation group

 Calling Sequence AreConjugate( a, b, G ) Conjugator( a, b, G )

Parameters

 G - a permutation group a - permutation b - permutation

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\mathrm{Perm}\left(\left[\left[2,4,6\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,5\right],\left[2,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,4\right],\left[2,5\right],\left[3,6\right]\right]\right)\right)$
 ${G}{≔}⟨\left({2}{,}{4}{,}{6}\right){,}\left({1}{,}{5}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right)⟩$ (1)
 > $a≔\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right],\left[5,6\right]\right]\right)$
 ${a}{≔}\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)$ (2)
 > $b≔\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,6\right],\left[4,5\right]\right]\right)$
 ${b}{≔}\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right)$ (3)
 > $\mathrm{AreConjugate}\left(a,b,G\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{AreConjugate}\left(a,b,\mathrm{Symm}\left(6\right)\right)$
 ${\mathrm{true}}$ (5)
 > $c≔\mathrm{Perm}\left(\left[\left[1,4\right],\left[2,5\right],\left[3,6\right]\right]\right)$
 ${c}{≔}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right)$ (6)
 > $\mathrm{AreConjugate}\left(a,c,G\right)$
 ${\mathrm{true}}$ (7)
 > $d≔\mathrm{Conjugator}\left(a,c,G\right)$
 ${d}{≔}\left({2}{,}{4}{,}{6}\right)$ (8)
 > ${d}^{-1}·a·d=c$
 $\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right){=}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right)$ (9)

Compatibility

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