Usually the best way to calculate in a group of automorphisms is to go
translate all calculations to an isomorphic group in a representation, for
which better algorithms are available, say a permutation group. This
translation can be done automatically using a NiceMonomorphism
(see NiceMonomorphism.)
Once a group knows to be a group of automorphisms (this can be achieved
by testing or setting the property IsGroupOfAutomorphisms
(see IsGroupOfAutomorphisms),
GAP will try itself to find such a nice monomorphism once calculations in
the automorphism group are done.
AssignNiceMonomorphismAutomorphismGroup( autgrp, group ) F
computes a nice monomorphism for autgroup acting on group and stores
it as NiceMonomorphism in autgrp.
If the centre of AutomorphismDomain of autgrp is trivial, the
operation will first try to represent all automorphisms by conjugation
(in group or a natural parent of group).
If this fails the operation tries to find a small subset of group on which the action will be faithful.
The operation sets the attribute NiceMonomorphism and does not return
a value.
If a good domain for a faithful permutation action is know already, a
homomorphism for the action on it can be created using
NiceMonomorphismAutomGroup. It might be stored by SetNiceMonomorphism
(see NiceMonomorphism).
NiceMonomorphismAutomGroup( autgrp, elms, elmsgens ) F
This function creates a monomorphism for an automorphism group
autgrp of a group by permuting the group elements in the list elms.
This list must be chosen to yield a faithful representation. elmsgens
is a list of generators which are a subset of elms. (They can differ
from the groups original generators.) It does not yet assign it as
NiceMonomorphism.
Another nice way of representing automorphisms as permutations has been described in Sims97. It it not yet available in GAP, a description however can be found in section Stabilizer Chains for Automorphisms Acting on Enumerators of ``Extending GAP''.
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GAP 4 manual