21.25 Permutation Group Records

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:

always true.

isFinite:

always 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:

List of orbitpoints of orbit[1] (which is the basepoint) under the action of the generators.

transversal:

Factorized inverse transversal as defined in Stabilizer Chains.

stabilizer:

Record for the stabilizer of the point 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:

A record, that contains information about creation of the stabilizer chain. For example, whether it has been computed using random methods (see Random Methods for Permutation Groups). Some functions also use this record for passing local information about basechanges.

stabChain:

A record, that contains all information about the stabilizer chain. Functions acessing the stabilizer chain should do it using this record, as it is planned to remove the above three components from the group record in the future. The components of the stabChain record are described below.

The components of the stabChain record for a group G are

identity:

Contains G.identity.

generators:

Contains a copy of the generators of G, created by ShallowCopy(G.generators).

orbit:

is the same as G.orbit.

transversal:

is the same as G.transversal.

stabilizer:

is the same as G.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.

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