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simultaneous conjugacy in Sn

asked 2023-02-13 16:20:54 +0200

matrixman gravatar image

Is there a built-in function in sage to determine whether two pairs (a,b) and (c,d) of elements of S_n (symmetric group of n letters) are simultaneously conjugate, i.e. checking whether there exists x such that

xa=cx and xb=dx ?

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answered 2023-02-13 20:59:11 +0200

Max Alekseyev gravatar image

updated 2023-02-13 21:00:52 +0200

Using GAP interface and the idea from here:

G=SymmetricGroup(3); a=G((1,2)); c=G((2,3))
b, d = c, a    # just as an example

C = (G(x) for x in gap.AsList( gap.RightCoset(gap.Centralizer(G,c), gap.RepresentativeAction(G,c,a)) ))
try:
    x = next(x for x in C if x*b==d*x)
    print('x=',x)
except StopIteration:
    print('No x exists!')
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Asked: 2023-02-13 16:20:54 +0200

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Last updated: Feb 13 '23