[GAP Forum] generators of fp groups

[Will Chen]

Dear all,

Suppose F is a finitely presented group, G a finite index subgroup. Often
the value of GeneratorsOfGroup(G) is far from minimal. I have 3 questions.

My first very vague/general question is - is it typically beneficial to
first reduce the number of generators of G before doing further
computations involving G?

Second, I have a question about trying to understand GAP's output when
using IsomorphismFpGroup. In my case I have a group G, which is a finite
index subgroup of a certain finitely presented group with generators "E"
and "V".

gap> G := GDATA[1870].stab;

Group(<194 generators>)

gap> GeneratorsOfGroup(G);

[ V^-1*E^2*V^-2*E*V*E^-2*V^2*E^-1, (E*V^-1)^4,
(E*V^-2*E)^2*V*E^-2*V*(V*E^-1)^2,
(V^-1*E)^2*E*V^-2*E*(V*E^-1)^2*E^-1*V^2*E^-1,
E*V^-1*(E*V^-2*E)^2*V*E^-2*V*(V*E^-1)^3, ***omitted the rest of the
output***


Next, I try to reduce the number of generators of G via IsomorphismFpGroup:


gap> f := IsomorphismFpGroup(G);

[ <[ [ 1, 1 ] ]|(V*E)^2*(V*E^-1)^2>, <[ [ 2, 1 ]
]|E*V*(E*V^-1)^2*E*V*E^-1>, <[ [ 3, 1 ] ]|(E*V)^3*E*(V^-1*E^-1)^2>, <[ [ 4,
1 ] ]|V*E*V^-1*(E*V)^3*(E^-1*V)^2>,

  <[ [ 5, 1 ] ]|(V^-1*E*V*E)^2*V*E^-1*V^-1*E^-1*V>, <[ [ 6, 1 ]
]|(V*E*V^-1*E)^2*V*E*V^-1*(E^-1*V)^2>, <[ [ 7, 1 ]
]|V*(E*V^-1)^2*(E*V)^2*E*V^-1*(E^-1*V)^2*E^-1*V^-1>,

  <[ [ 8, 1 ] ]|V*(E*V^-1)^5*(E^-1*V^-1)^2*E^-1*V>, <[ [ 9, 1 ]
]|V^-1*(E*V)^2*(E*V^-1)^3*E^-1*V^-1*(E^-1*V)^2>, <[ [ 10, 1 ]
]|V*(E*V^-1)^2*(E*V^-1*E*V)^2*(E^-1*V^-1)^2*E^-1*V> ] ->

[ F1, F2, F3, F4, F5, F6, F7, F8, F9, F10 ]


What are these "[1, 1]", "[2, 1]" that are appearing? They continue to
appear even if I access the corresponding element:


gap> finv := InverseGeneralMapping(f);;

gap> Image(f).1^finv;

<[ [ 1, 1 ] ]|(V*E)^2*(V*E^-1)^2>

gap>

After reading the GAP manual they seem to have something to do with
"straight line program elements", but I still don't know how to interpret
them. What is the meaning of [a,b]?

My third and final question is - is it possible to update the attribute
"GeneratorsOfGroup" for an existing group? For example, I'd like to make it
so that the group "G" above uses the 10 generators obtained via
IsomorphismFpGroup as opposed to the default "194 generators".

Thanks,

- Will


-- 

William Chen
Member, School of Mathematics
Institute for Advanced Study
Princeton, NJ, 08540
oxeimon at gmail.com