This chapter describes some pre-defined group classes, namely the classes of all abelian, nilpotent, and supersoluble groups. Moreover, there are some functions constructing the classes of all p-groups, π-groups, and abelian groups whose exponent divides a given positive integer.
The definitions of these group classes can also serve as further examples of how group classes can be defined using the methods described in the preceding chapters.
The global variable
TrivialGroups contains the class of all trivial groups. It is a
subgroup closed saturated Fitting formation.
This global variable contains the class of all finite nilpotent groups. It is a subgroup closed saturated Fitting formation.
This global variable contains the class of all finite supersoluble groups. It is a subgroup closed saturated formation.
is the class of all abelian groups. It is a subgroup closed formation.
returns the class of all abelian groups of exponent dividing n, where n is a positive integer. It is always a subgroup-closed formation.
constructs the class of all pi-groups. pi may be a non-empty class or a set of primes. The result is a subgroup-closed saturated Fitting formation.
returns the class of all p-groups, where p is a prime. The result is a subgroup-closed saturated Fitting formation.
This function returns a projector for the class of all finite nilpotent groups. For a definition, see Projector. Note that the nilpotent projectors of a finite soluble group equal its a Carter subgroups, that is, its self-normalizing nilpotent subgroups.
These functions return a projector for the class of all finite supersoluble groups. For a definition, see Projector.
is the set of all (integral) primes. This should be
installed as value for
) if the group class
grpclass contains cyclic groups of prime order p for arbitrary primes p.
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