[GAP Forum] Responding Time (Group Homomorphisms by Images)

[Minghui Liu]

Dear Sir/Madam,

I am trying to use GAP to construct group homomorphisms. In fact, I have a
list of groups K_n, each K_n is generated by n elements x_1, ... x_n
subject to some relations where n is at least 2. Now I want to construct
group homomorphisms f_{n,k} from K_n to K_{n-1} defined by f_{n,k}(x_1) =
x_1, ..., f_{n,k}(x_{k-1}) = x_{k-1}, f_{n,k}(x_k) = e, f_{n,k}(x_{k+1}) =
x_{k}, ..., f_{n,k}(x_n) = x_{n-1}.

For example,

f_{3,1} from K_3 to K_2 maps x_1 to e, x_2 to x_1 and x_3 to x_2;

f_{3,2} from K_3 to K_2 maps x_1 to x_1, x_2 to e and x_3 to x_2;

f_{3,3} from K_3 to K_2 maps x_1 to x_1, x_2 to x_2 and x_3 to e.

Now I am interested in studying the intersection of ker(f_{n, k}), where n
is some fixed integer and k runs from 1 to n. When I put my question to
GAP, I have used the following commands:

F2:=FreeGroup("a","b");

B2:=[Comm(Comm(F2.1,F2.2),F2.1),Comm(Comm(F2.1,F2.2),F2.2)];

K2:=F2/B2;

F3:=FreeGroup("a","b","c");

B3:=[Comm(Comm(F3.1,F3.2),F3.1),Comm(Comm(F3.1,F3.2),F3.2),
Comm(Comm(F3.1,F3.3),F3.1),Comm(Comm(F3.1,F3.3),F3.3),
Comm(Comm(F3.2,F3.1),F3.2),Comm(Comm(F3.2,F3.1),F3.1),
Comm(Comm(F3.2,F3.3),F3.2),Comm(Comm(F3.2,F3.3),F3.3),
Comm(Comm(F3.3,F3.1),F3.3),Comm(Comm(F3.3,F3.1),F3.1),
Comm(Comm(F3.3,F3.2),F3.3),Comm(Comm(F3.3,F3.2),F3.2),
Comm(Comm(Comm(F3.1,F3.2),F3.3),F3.1),
Comm(Comm(Comm(F3.1,F3.2),F3.3),F3.2),
Comm(Comm(Comm(F3.1,F3.2),F3.3),F3.3),
Comm(Comm(Comm(F3.1,F3.3),F3.2),F3.1),
Comm(Comm(Comm(F3.1,F3.3),F3.2),F3.2),
Comm(Comm(Comm(F3.1,F3.3),F3.2),F3.3),
Comm(Comm(Comm(F3.2,F3.1),F3.3),F3.1),
Comm(Comm(Comm(F3.2,F3.1),F3.3),F3.2),
Comm(Comm(Comm(F3.2,F3.1),F3.3),F3.3),
Comm(Comm(Comm(F3.2,F3.3),F3.1),F3.1),
Comm(Comm(Comm(F3.2,F3.3),F3.1),F3.2),
Comm(Comm(Comm(F3.2,F3.3),F3.1),F3.3),
Comm(Comm(Comm(F3.3,F3.1),F3.2),F3.1),
Comm(Comm(Comm(F3.3,F3.1),F3.2),F3.2),
Comm(Comm(Comm(F3.3,F3.1),F3.2),F3.3),
Comm(Comm(Comm(F3.3,F3.2),F3.1),F3.1),
Comm(Comm(Comm(F3.3,F3.2),F3.1),F3.2),
Comm(Comm(Comm(F3.3,F3.2),F3.1),F3.3)];

K3:=F3/B3;

f321:=GroupHomomorphismByImages(K3,K2,[K3.1,K3.2,K3.3],[One(K2),K2.1,K2.2]);

Then it takes like forever for my GAP to respond. Thus it seems hopeless to
deal with the cases where n is 4 or larger. Just in case that you need this
information, I am running GAP4.6.5 on my six-year-old computer.

Any comments or suggestions? Your assistance will be greatly appreciated.

By the way, after hitting "Enter" how can we roughly know how long it would
take for GAP to respond? From my experience sometimes it could be a few
minutes and sometimes it does not respond after about one hour and I choose
to terminate it. If I know it would respond in say two hours I probably can
wait; but what if it takes dozens or hundreds of hours?

Best regards,

Minghui