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The PolycyclicGroup-constructor allows complete flexibility
in defining a pc-group. However, it is often more convenient to
have Magma compute a pc-presentation based on some other description
of the group. The PCGroup function will produce a
pc-presentation for a finite group in various categories such as
GrpPerm and GrpMat. Converting from a FINITELY PRESENTED GROUPS
group is trickier, since the original group need not be finite.
There are two functions provided to produce pc-presentations for
certain quotients of finitely-presented groups. The pQuotient
function constructs a pc-presentation for the largest p-group
quotient having specified lower exponent-p class. Similarly,
SolubleQuotient will compute the largest soluble quotient
subject to certain restrictions.
Each of these functions also provides a homomorphism (isomorphism in the
case of PCGroup) from the original group to the new pc-group.
More information on each of
the two quotient functions can be found in Chapter FINITELY PRESENTED GROUPS.
PCGroup(G) : GrpMat -> GrpPC, Map
[Future release] PCGroup(G) : GrpFP -> GrpPC
A GrpPC representation of the group G and the isomorphism.
Workspace: RngIntElt Default: 1000000
Metabelian: BoolElt Default: false
Exponent: RngIntElt Default: 0
Print: RngIntElt Default: 0
Given a finitely presented group F, a prime p, and a positive
integer c, this function constructs a consistent power-conjugate
presentation for the largest p-quotient H of F having
lower exponent-p class at most c. If c is given as zero,
then the limit 127 is placed on the class.
The function returns both the p-quotient H defined by a pc-presentation
and the homomorphism from F to H.
SolvableQuotient(G) : Grp -> GrpPC, Map
A GrpPC representation P of the largest solvable quotient of G and
the homomorphism φ: G -> P.
We use PCGroup to produce a pc-presentation for
a matrix group.
> GL := GeneralLinearGroup(4,GF(3));
> S3 := Sylow(GL,3);
> P := PCGroup(S3);
> P;
GrpPC : P of order 729 = 3^6
PC-Relations:
P.2^P.1 = P.2 * P.4^2,
P.3^P.1 = P.3 * P.5^2,
P.3^P.2 = P.3 * P.6^2,
P.5^P.2 = P.4 * P.5,
P.6^P.1 = P.4 * P.6
Given a pc-group, it is straight-forward to convert it to
a FINITELY PRESENTED GROUPS or GrpGPC representation by
using the appropriate transfer function. If one wishes
to have a permutation representation of the group, this
requires more cleverness. The CosetAction function
can be used to compute the permutation representation of
a group on a subgroup. If the subgroup is chosen to
have trivial core, then the permutation group obtained
will be isomorphic to the original group.
Given an abelian pc-group G, return a GrpAb group H
isomorphic to G and an isomorphism φ: G -> H.
A FINITELY PRESENTED GROUPS representation F of G and the isomorphism from G to F.
A GrpGPC representation F of G and the isomorphism
from G to F.
Take one of the groups of order 2 6 * 3 2.
> G := SmallGroup(576, 4123);
> G;
GrpPC : G of order 576 = 2^6 * 3^2
PC-Relations:
G.1^2 = Id(G),
G.2^2 = Id(G),
G.3^2 = G.5,
G.4^3 = Id(G),
G.5^2 = G.7,
G.6^2 = G.7,
G.7^2 = Id(G),
G.8^3 = Id(G),
G.2^G.1 = G.2 * G.6,
G.6^G.1 = G.6 * G.7,
G.6^G.2 = G.6 * G.7,
G.8^G.1 = G.8^2
Since G is small, we can search for a minimum degree
permutation presentation by brute force. First we build
a set containing all the subgroups.
> SL := Subgroups(G);
> T := {X`subgroup: X in SL};
> #T;
243
Then, we select those subgroups with trivial core, and
find one with the smallest index.
> TrivCore := {H:H in T| #Core(G,H) eq 1};
> mdeg := Min({Index(G,H):H in TrivCore});
> Good := {H: H in TrivCore| Index(G,H) eq mdeg};
> #Good;
3
> H := Rep(Good);
We then use CosetAction to construct the permutation representation
on the cosets of H.
> f,P,K := CosetAction(G,H);
> #K;
1
> IsPrimitive(P);
false
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