[GAP Forum] Encode Homomorphism Between Groups
Jeffrey Rolland
rollandj at uwm.edu
Sat Jan 23 21:05:15 GMT 2010
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Hello, all!
I am a GAP newbie. I am trying to create a semidirect product in GAP,
and I am having a little trouble encoding the homomorphism.
The semidirect product group G is Q |x S, where Q = Z, the integers
(i.e., the free group on 1 generator), P = SL(2,5), and S = P_1*P_2*P_3
(three copies of P). (I have successfully coded the groups into GAP.)
Now, the matrices A = [[0 1][4 0]] and B = [[0 1][4 1]] (row vectors) in
SL(2,5) have the property that \Xi(A) is not equal to B or B^{-1} for
any \Xi \in Aut(P). (This result can be proven using the Jordan normal
forms of A and B, or verified directly using Maple, e.g., and the fact
that Aut(P) is isomorphic to PGL(2,5).)
Let A_2 be the copy of A in P_2 and A_3 be the copy of A in P_3, and
similarly for B_@ and B_3.
An automorphism \phi of S can be defined by setting \phi(X_1) =
(A_1A_2)^{-1}X_1(A_1A_2) for X_! in P_1 and \phi(X_2) = X_2 and
\phi(X_3) = X_3 for X_2 \in P_2 and X_3 \in P_3.
Similarly, an automorphism \psi of S can be defined by setting \psi(X_1)
= (B_1B_2)^{-1}X_1(B_1B_2) for X_! in P_1 and \psi(X_2) = X_2 and
\psi(X_3) = X_3 for X_2 \in P_2 and X_3 \in P_3.
Now, I wish to create a homomorphism \Phi: Q -> Aut(S) by sending
\Phi(q) = \phi^q (\phi composed with itself q times) and \Psi: Q -> S by
sending \Psi(q) = \psi^q (\psi composed with itself q times).
Note that S is a finitely presented group with GAP generators
gap> GeneratorsOfGroup(S);
[ f1, f2, f3, f4, f5, f6, f7, f8, f9 ]
and relators
gap> RelatorsOfFpGroup(S);
[ f1^2*f3, f2^3*f3, f1*f2*f1*f2*f1*f2*f1*f2*f1*f2, f3^-1*f1^-1*f3*f1,
f3^-1*f2^-1*f3*f2, f3^2, f4^2*f6, f5^3*f6, f4*f5*f4*f5*f4*f5*f4*f5*f4*f5,
f6^-1*f4^-1*f6*f4, f6^-1*f5^-1*f6*f5, f6^2, f7^2*f9, f8^3*f9,
f7*f8*f7*f8*f7*f8*f7*f8*f7*f8, f9^-1*f7^-1*f9*f7, f9^-1*f8^-1*f9*f8,
f9^2 ]
where f1 = [[2 0][0 3]] \in P_1, f2 = [[4 1][4 0]] \in P_1, and f3 =
[[-1 0][0 -1]] \in P_1, and similarly for f4-f6 and f7-f9 (i.e., f4 =
[[2 0][0 3]] \in P_2, f5 = [[4 1][4 0]] \in P_2, etc.).
I have created a program to write each of A and B in terms of f1 and f2
(and similarly for f4 and f5 and f7 and f8) (the generators f3, f6, and
f9 are somewhat superfluous). The results are A =
f2*f1*f2*f2*f1*f2*f1*f2*f2*f1*f2 and B = f1*f1*f2*f2.
So, could someone pretty-please help me to encode the homomorphisms Phi
and Psi so that I can create the semidirect products
G_1:=SemidirectProduct(Q,Phi,S) and G_2:=SemidirectProduct(Q,Psi,S)?
Many thanks in advance.
Sincerely,
- --
Jeffrey Rolland
<rollandj at uwm.edu>
P.S. Attached is a GAP file with my work to date.
JJR
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