Basic Group Properties

Contents

Infrastructure

The functions described here provide access to basic information stored for a pc-group G.

G . i : GrpPC, RngIntElt -> GrpPCElt
The i-th pc-generator for G. A negative subscript indicates that the inverse of the generator is to be created. G.0 is Identity(G).
Generators(G) : GrpPC -> SetEnum
A set containing the defining generators for G. If G is a p-group, this is guaranteed to be a minimal set of generators. For non-p-groups, this will be the set of pc-generators.
NumberOfGenerators(G) : GrpPC -> RngIntElt
Ngens(G) : GrpPC -> RngIntElt
The number of defining generators for G.
PCGenerators(G) : GrpPC -> SetIndx
An indexed set containing the pc-generators for G.
NumberOfPCGenerators(G) : GrpPC -> RngIntElt
NPCGenerators(G) : GrpPC -> RngIntElt
NPCgens(G) : GrpPC -> RngIntElt
The number of pc-generators for G.
PCPrimes(G) : GrpPC -> [RngIntElt]
A sequence [p1, ..., pn] containing the primes associated with the pc-generators of G. The i-th term of the sequence contains the prime associated with generator ai of G for i = 1, ..., n.

Numerical Invariants

Magma has built-in functions to compute the order and exponent of a group.

Order(G) : GrpPC -> RngIntElt
# G : GrpPC -> RngIntElt
The order of the group G, returned as an ordinary integer.
FactoredOrder(G) : GrpPC -> [<RngIntElt, RngIntElt>]
The factored order of the group G.
Exponent(G) : GrpPC -> RngIntElt
The exponent of the group G.

Predicates

Magma has built-in functions to check standard group properties.

IsAbelian(G) : GrpPC -> BoolElt
Returns true if the group G is abelian, false otherwise.
IsCyclic(G) : GrpPC -> BoolElt
Returns true if the group G is cyclic, false otherwise.
IsElementaryAbelian(G) : GrpPC -> BoolElt
Returns true if the group G is elementary abelian, false otherwise.
IsNilpotent(G) : GrpPC -> BoolElt
Returns true if the group G is nilpotent, false otherwise.
IsPerfect(G) : GrpPC -> BoolElt
Returns true if the group G is perfect, false otherwise. A soluble group G is perfect only if it is trivial.
IsSimple(G) : GrpPC -> BoolElt
Returns true if the group G is simple, false otherwise.
IsSoluble(G) : GrpPC -> BoolElt
IsSolvable(G) : GrpPC -> BoolElt
Returns true if the group G is soluble, false otherwise. It always returns the value true for a pc-group.
IsTrivial(G) : GrpPC -> BoolElt
Returns true if the group G has order 1, false otherwise.
IsSpecial(G) : GrpPC -> BoolElt
Given a p-group G, return true if G is special, false otherwise.
IsExtraSpecial(G) : GrpPC -> BoolElt
Given a p-group G, return true if G is extra-special, false otherwise.

Example GrpPC_group-props (H69E4)

We use a presentation to define an extraspecial 3-group of exponent 9. 
> E := PolycyclicGroup<a1,a2,b1,b2,z|a1^3,a2^3,b1^3=z,b2^3=z,
>   z^3,b1^a1=b1*z,b2^a2=b2*z>;
The sequence of base, exponent pairs from FactoredOrder shows us that the group has order 35. 
> FactoredOrder(E);
[ <3, 5> ]
> Exponent(E);
9
As well as with the Order function, one can get the size of a group by using the # shorthand. 
> D3 := DihedralGroup(GrpPC, 3);
> #D3;
6
> IsNilpotent(D3);
false
V2.28, 13 July 2023