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6 Examples of group classes

Sections

  1. Pre-defined group classes
  2. Pre-defined projector functions
  3. Pre-defined sets of primes

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.

6.1 Pre-defined group classes

  • TrivialGroups V

    The global variable TrivialGroups contains the class of all trivial groups. It is a subgroup closed saturated Fitting formation.

  • NilpotentGroups V

    This global variable contains the class of all finite nilpotent groups. It is a subgroup closed saturated Fitting formation.

  • SupersolubleGroups V
  • SupersolvableGroups V

    This global variable contains the class of all finite supersoluble groups. It is a subgroup closed saturated formation.

  • AbelianGroups V

    is the class of all abelian groups. It is a subgroup closed formation.

  • AbelianGroupsOfExponent(n) F

    returns the class of all abelian groups of exponent dividing n, where n is a positive integer. It is always a subgroup-closed formation.

  • PiGroups(pi) F

    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.

  • PGroups(p) F

    returns the class of all p-groups, where p is a prime. The result is a subgroup-closed saturated Fitting formation.

    6.2 Pre-defined projector functions

  • NilpotentProjector(grp) A

    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.

  • SupersolubleProjector(grp) A
  • SupersolvableProjector(grp) A

    These functions return a projector for the class of all finite supersoluble groups. For a definition, see Projector.

    6.3 Pre-defined sets of primes

  • AllPrimes V

    labelAllPrimesrelax is the set of all (integral) primes. This should be installed as value for Characteristic(grpclass) if the group class grpclass contains cyclic groups of prime order p for arbitrary primes p.

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    CRISP manual
    March 2016