Basic Group Properties

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 [\(p_1,\ldots,p_n\)] containing the primes associated with the pc-generators of \(G\). The \(i\)-th term of the sequence contains the prime associated with generator \(a_i\) of \(G\) for \(i = 1,\ldots,n\).

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: Basic Properties of Pc-Groups (ex-e86fe2)

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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 \(3^5\).

> 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

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