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The functions described here provide access to basic information stored for a
pc-group G.
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).
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.
Ngens(G) : GrpPC -> RngIntElt
The number of defining generators for G.
An indexed set containing the pc-generators for G.
NPCGenerators(G) : GrpPC -> RngIntElt
NPCgens(G) : GrpPC -> RngIntElt
The number of pc-generators for G.
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.
Magma has built-in functions to compute the order
and exponent of a group.
# G : GrpPC -> RngIntElt
The order of the group G, returned as an ordinary integer.
The factored order of the group G.
The exponent of the group G.
Magma has built-in functions
to check standard group properties.
Returns true if the group G is abelian, false otherwise.
Returns true if the group G is cyclic, false otherwise.
Returns true if the group G is elementary abelian, false otherwise.
Returns true if the group G is nilpotent, false otherwise.
Returns true if the group G is perfect, false otherwise.
A soluble group G is perfect only if it is trivial.
Returns true if the group G is simple, false otherwise.
IsSolvable(G) : GrpPC -> BoolElt
Returns true if the group G is soluble, false otherwise.
It always returns the value true for a pc-group.
Returns true if the group G has order 1, false otherwise.
Given a p-group G, return true if G is special, false otherwise.
Given a p-group G, return true if G is extra-special, false otherwise.
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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