7.78 Subgroup Conjugacy Class Records

Each conjugacy class of subgroups C is represented as a record with at least the following components.

isDomain:

always true, because conjugacy classes of subgroups are domains.

isConjugacyClassSubgroups:

as well, this entry is always set to true.

group:

The group in which the members of this conjugacy class lie. This is not necessarily a parent group; it may also be a subgroup.

representative:

The representative of the conjugacy class of subgroups as domain.

The following components are optional and may be bound by some functions which compute or make use of their value.

normalizer:

The normalizer of C.representative in C.group.

normalizerLattice:

A special entry that is used when the conjugacy classes of subgroups are computed by ConjugacyClassesSubgroups. It determines the normalizer of the subgroup C.representative. It is a list of length 2. The first element is another conjugacy class D (in the same group), the second is an element g in C.group. The normalizer of C.representative is then D.representative ^ g.

conjugands:

A right transversal of the normalizer of C.representative in C.group. Thus the elements of the class C can be computed by conjugating C.representative with those elements.

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