All groups are represented by a record that contains information about the group. A permutation group record contains the following components in addition to those described in section Group Records.
isPermGroup
:true
.
isFinite
:true
as permutation groups are always of finite order.
A stabilizer chain (see Stabilizer Chains) is stored recursively in GAP. The group record of a permutation group G with a stabilizer chain has the following additional components.
orbit
:orbit[1]
(which is the basepoint) under
the action of the generators.
transversal
:
stabilizer
:orbit[1]
in the group
generated by generators
. The components of this record are
again generators
, orbit
, transversal
and stabilizer
. The
last stabilizer in the stabilizer chain only contains the
component generators
, which is an empty list.
stabChainOptions
:
stabChain
:stabChain
record are described below.
The components of the stabChain
record for a group G are
identity
:.identity
.
generators
:ShallowCopy(G.generators)
.
orbit
:.orbit
.
transversal
:.transversal
.
stabilizer
:.stabilizer
.
Note that the values of all these components are changed by functions that
change, extend, or reduce a base (see MakeStabChain, ExtendStabChain,
and ReduceStabChain).
Note that the records that represent the stabilizers are not themselves
group records (see Group Records). Thus you cannot take such a
stabilizer and apply group functions to it. The last stabilizer
in the
stabilizer chain is a record whose component generators
is empty.
GAP 3.4.4