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finding conjugating element for two conjugate permutations

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

matrixman gravatar image

updated 2023-02-13 20:40:21 +0200

Max Alekseyev gravatar image

Suppose a and b are elements of a symmetric group that are conjugate, how to find a conjugating element in sage?

For example,

G=SymmetricGroup(3);
a=G((1,2))
b=G((2,3))

Then

G.conjugacy_class(a) == G.conjugacy_class(b)

is true, but is there a function to give x such that xa==bx?

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I don't know of anything built in, but how large a group are you working with? If I use G=SymmetricGroup(11), a naïve brute force search for a single element x takes 7µs on my machine, and a brute force search for all elements x takes about 30 seconds.

John Palmieri gravatar imageJohn Palmieri ( 2023-02-13 20:03:40 +0200 )edit

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answered 2023-02-13 20:37:47 +0200

Max Alekseyev gravatar image

updated 2023-02-14 15:31:30 +0200

This can be done using interface to GAP:

G=SymmetricGroup(3); a=G((1,2)); b=G((2,3))
x = G( gap.RepresentativeAction(G,b,a) )
print( x*a == b*x )
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Max Alekseyev gravatar imageMax Alekseyev ( 2024-02-10 21:04:14 +0200 )edit

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Asked: 2023-02-13 16:13:16 +0200

Seen: 66 times

Last updated: Feb 14 '23