36.14 Group Properties

A group is cyclic if it can be generated by one element. For a cyclic group, one can compute a generating set consisting of only one element using MinimalGeneratingSet (see MinimalGeneratingSet).

A group G is elementary abelian if it is commutative and if there is a prime p such that the order of each element in G divides p.

A group is nilpotent if the lower central series (see LowerCentralSeriesOfGroup for a definition) reaches the trivial subgroup in a finite number of steps.

A group is perfect if it equals its derived subgroup (see DerivedSubgroup).

A group is solvable if the derived series (see DerivedSeriesOfGroup for a definition) reaches the trivial subgroup in a finite number of steps.

For finite groups this is the same as being polycyclic (see IsPolycyclicGroup), and each polycyclic group is solvable, but there are infinite solvable groups that are not polycyclic.

A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups this is the same as if the group is solvable (see IsSolvableGroup).

A finite group is supersolvable if it has a normal series with cyclic factors.

A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.

A group is simple if it has no nontrivial normal subgroups.

For a finite simple group G, IsomorphismTypeFiniteSimpleGroup returns a record with components series, name and possibly parameter, describing the isomorphism type of G. The component name is a string that gives name(s) for G, and series is a string that describes the following series.

(If different characterizations of G are possible only one is given by series and parameter, while name may give several names.)

"A"

Alternating groups, parameter gives the natural degree.

"L"

Linear groups (Chevalley type A), parameter is a list [n,q] that indicates L(n,q).

"2A"

Twisted Chevalley type ²A, parameter is a list [n,q] that indicates ²A(n,q).

"B"

Chevalley type B, parameter is a list [n,q] that indicates B(n,q).

"2B"

Twisted Chevalley type ²B, parameter is a value q that indicates ²B(2,q).

"C"

Chevalley type C, parameter is a list [n,q] that indicates C(n,q).

"D"

Chevalley type D, parameter is a list [n,q] that indicates D(n,q).

"2D"

Twisted Chevalley type ²D, parameter is a list [n,q] that indicates ²D(n,q).

"3D"

Twisted Chevalley type ³D, parameter is a value q that indicates ³D(4,q).

"E"

Exceptional Chevalley type E, parameter is a list [n,q] that indicates E_n(q). The value of n is 6,7 or 8.

"2E"

Twisted exceptional Chevalley type E₆, parameter is a value q that indicates ²E₆(q).

"F"

Exceptional Chevalley type F, parameter is a value q that indicates F(4,q).

"2F"

Twisted exceptional Chevalley type ²F (Ree groups), parameter is a value q that indicates ²F(4,q).

"G"

Exceptional Chevalley type G, parameter is a value q that indicates G(2,q).

"2G"

Twisted exceptional Chevalley type ²G (Ree groups), parameter is a value q that indicates ²G(2,q).

"Spor"

Sporadic groups, name gives the name.

"Z"

Cyclic groups of prime size, parameter gives the size.

An equal sign in the name denotes different naming schemes for the same group, a tilde sign abstract isomorphisms between groups constructed in a different way.

gap> IsomorphismTypeFiniteSimpleGroup(Group((4,5)(6,7),(1,2,4)(3,5,6)));
rec( series := "L", parameter := [ 2, 7 ], name:=
  "A(1,7)=L(2,7)~B(1,7)=O(3,7)~C(1,7)=S(2,7)~2A(1,7)=U(2,7)~A(2,2)=L(3,2)")

tests whether the group G can be generated by a finite number of generators. (This property is mainly used to obtain finiteness conditions.)

A group is called locally finite if every finitely generated subgroup is finite. This property checks whether the group U is a subset of a locally finite group. This is used to check whether finite generation will imply finiteness, as it does for example for permutation groups.

A p-group is a finite group whose order (see Size) is of the form pⁿ for a prime integer p and a nonnegative integer n. IsPGroup returns true if G is a p-group, and false otherwise.

If G is a nontrivial p-group (see IsPGroup), PrimePGroup returns the prime integer p; if G is trivial then PrimePGroup returns fail. Otherwise an error is issued.

The p-class of a p-group G (see IsPGroup) is the length of the lower p-central series (see PCentralSeries) of G. If G is not a p-group then an error is issued.

For a p-group G (see IsPGroup), RankPGroup returns the rank of G, which is defined as the minimal size of a generating system of G. If G is not a p-group then an error is issued.

gap> h:=Group((1,2,3,4),(1,3));;
gap> PClassPGroup(h);
2
gap> RankPGroup(h);
2

Note that the following following functions, although they are mathematical properties, are not properties in the sense of GAP (see Attributes and Properties), as they depend on a parameter.

A group is p-solvable if every chief factor is either not divisible by p or solvable.

@Currently no method is installed!@

A group is p-nilpotent if it possesses a normal p-complement.

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