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I would like to know how to obtain the conjugacy class of some tuple in GAP, or how two know if two tuples are conjugate by an element of some group (permutation group, more especifically).

I know that for unidimensional cases I can use IsConjugate or ConjugacyClass, but I don't know how to do for a list of elements.

MaríaCC
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1 Answers1

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To test in GAP whether two tuples are conjugate, first test whether the first entries are conjugate, then map the whole tuple with a conjugator and test for conjugacy of the second component under the centralizer of the first. This is implemented under RepresentativeAction:

RepresentativeAction(group,tuple1,tuple2,OnTuples);

e.g.

gap> g:=SymmetricGroup(10);;
gap> tup:=[ (1,2,5,4,10,3,6,8)(7,9), (1,3,2,10,6,5)(4,9,7), (1,7,2,10,3,4,6)(5,9,8) ];;
gap> tup2:=[ (1,3,5,6,7,9,10,2)(4,8), (1,6,9,3,2,7)(4,8,5), (1,6,7,5,9,2,8)(3,4,10) ];;
gap> RepresentativeAction(g,tup,tup2,OnTuples);
(1,2)(3,7,8,10,6,9,4,5)
ahulpke
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  • Thanks! So afeter cheking the firs component I just need to see if RepresentativeAction(g,t1,t1,OnTuples)=fail. – MaríaCC Dec 01 '18 at 18:40
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    @MaríaCC You don't even need to check the first component, RepresentativeAction does both. – ahulpke Dec 01 '18 at 19:19