Dear Minghui,
You can proceed as follows:
gap> F := FreeGroup(4);
<free group on the generators [ f1, f2, f3, f4 ]>
gap> x := GeneratorsOfGroup(F);
[ f1, f2, f3, f4 ]
gap> S4 := SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> w := x[1]*x[2]*x[3]^-1*x[4]^-2; # your sample word
f1*f2*f3^-1*f4^-2
gap> rels := List(AsList(S4),g->MappedWord(w,x,Permuted(x,g))); # the 24
relators
[ f1*f2*f3^-1*f4^-2, f1*f2*f4^-1*f3^-2, f1*f3*f2^-1*f4^-2, f1*f4*f2^-1*f3^-2,
f1*f3*f4^-1*f2^-2, f1*f4*f3^-1*f2^-2, f2*f1*f3^-1*f4^-2, f2*f1*f4^-1*f3^-2,
f3*f1*f2^-1*f4^-2, f4*f1*f2^-1*f3^-2, f3*f1*f4^-1*f2^-2, f4*f1*f3^-1*f2^-2,
f2*f3*f1^-1*f4^-2, f2*f4*f1^-1*f3^-2, f3*f2*f1^-1*f4^-2, f4*f2*f1^-1*f3^-2,
f3*f4*f1^-1*f2^-2, f4*f3*f1^-1*f2^-2, f2*f3*f4^-1*f1^-2, f2*f4*f3^-1*f1^-2,
f3*f2*f4^-1*f1^-2, f4*f2*f3^-1*f1^-2, f3*f4*f2^-1*f1^-2, f4*f3*f2^-1*f1^-2 ]
Does this help you?
Best regards,
Stefan
-----------------------------------------------------------------------------
Dr. Stefan Kohl, https://stefan-kohl.github.io/
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________________________________
From: Minghui Liu <minghuiliu2...@gmail.com>
Sent: Saturday, March 23, 2019 10:02:48 PM
To: forum@gap-system.org
Subject: [GAP Forum] Symmetrising presented group from a given presentation
Dear GAP users,
I have the following question on inputting symmetrised relations in a group.
Let G be a group generated by x_1, x_2, x_3, x_4 with one relation for
example x_1*x_2*x_3^(-1)*x_4^(-2). Now I wish to let the symmetric group
S_4 on 4 letters acts on this relation and therefore obtain a total of
4!=24 relators. For example, one of the obtained relator is
x_2*x_3*x_4^(-1)*x_1^(-2). Is there any easy way for me to write all 24
relators in GAP?
Thank you!
Minghui
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