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A number of functions are provided which construct various standard groups.
The effect of these functions is to construct the group on some standard set
of generators.
The group category of the result may be specified as an argument to the
function.
AbelianGroup(Q) : [ RngIntElt ] -> GrpAb
Construct the abelian group defined by the sequence Q =
[n1, ..., nr] of positive integers. The function constructs
the direct product of cyclic groups Zn1 x Zn2 x ... x Znr. In some categories, ni may
also be 0, denoting the infinite cyclic group Z.
If the single-argument version of the function is used,
the group will be constructed in the category GrpAb;
otherwise, its category will be C,
where C may be GrpAb, GrpFP, GrpGPC, GrpPC or
GrpPerm.
AlternatingGroup(n) : RngIntElt -> GrpPerm
Alt(C, n) : Cat, RngIntElt -> GrpFin
Alt(n) : RngIntElt -> GrpPerm
Construct the alternating group on n letters.
If the single-argument version of the function is used,
the group will be constructed in the category GrpPerm;
otherwise, its category will be C,
where C may be GrpFP or GrpPerm.
CyclicGroup(n) : RngIntElt -> GrpPerm
Construct the cyclic group of order n.
If the single-argument version of the function is used,
the group will be constructed in the category GrpPerm;
otherwise, its category will be C,
where C may be GrpAb, GrpFP, GrpGPC, GrpPC or
GrpPerm.
DihedralGroup(n) : RngIntElt -> GrpPerm
Construct the dihedral group of order 2 * n.
If the single-argument version of the function is used,
the group will be constructed in the category GrpPerm;
otherwise, its category will be C,
where C may be GrpFP, GrpGPC, GrpPC or GrpPerm.
SymmetricGroup(n) : RngIntElt -> GrpPerm
Sym(GrpFin, n) : Cat, RngIntElt -> GrpFin
Sym(n) : Cat, RngIntElt -> GrpPerm
Construct the symmetric group on n letters.
If the single-argument version of the function is used,
the group will be constructed in the category GrpPerm;
otherwise, its category will be C,
where C may be GrpFP or GrpPerm.
ExtraSpecialGroup(p, n : parameters) : RngIntElt, RngIntElt -> GrpPerm
Given a prime p and a small positive integer n, construct an
extra-special group G of order p2n + 1. The isomorphism type of G
can be selected using the parameter Type described below.
If the two-argument version of the function is used, the group will be
constructed in the category GrpPerm; otherwise, its category will
be C, where C may be GrpFP, GrpGPC, GrpPC or
GrpPerm. If C is GrpFP, GrpPC or GrpPerm,
the prime p must be small.
Type: MonStgElt Default: "+"
Possible values for this parameter are "+" (default) and "-".
If Type is set to "+", the function returns for p = 2 the central
product of n copies of the dihedral group of order 8, and for p > 2
it returns the unique extra-special group of order p2n + 1 and exponent p.
If Type is set to "-", the function returns for p = 2 the central
product of a quaternion group of order 8 and n - 1 copies of the dihedral
group of order 8, and for p > 2 it returns the unique extra-special group
of order p2n + 1 and exponent p2.
- (1)
- The abelian group Z6 x Z2 x Z7
in the category GrpAb:
> A := AbelianGroup([6, 2, 7]);
> A;
Abelian Group isomorphic to Z/2 + Z/42
Defined on 3 generators
Relations:
6*A.1 = 0
2*A.2 = 0
7*A.3 = 0
- (2)
- The alternating group on 6 letters as a permutation group:
> A6 := Alt(6);
> A6;
Permutation group A6 acting on a set of cardinality 6
Order = 360 = 2^3 * 3^2 * 5
(1, 2)(3, 4, 5, 6)
(1, 2, 3)
- (3)
- The dihedral group of order 8 as a GrpPC:
> D8 := DihedralGroup(GrpPC, 4);
> D8;
GrpPC : D8 of order 8 = 2^3
PC-Relations:
D8.2^2 = D8.3,
D8.2^D8.1 = D8.2 * D8.3
- (4)
- The symmetric group on 7 letters as a finitely presented group
on generators a and b:
> S7<a, b> := SymmetricGroup(GrpFP, 7);
> S7;
Finitely presented group S7 on 2 generators
Relations
a^7 = Id(S7)
b^2 = Id(S7)
(a * b)^6 = Id(S7)
(a^-1 * b * a * b)^3 = Id(S7)
(b * a^-2 * b * a^2)^2 = Id(S7)
(b * a^-3 * b * a^3)^2 = Id(S7)
Given two groups G and H belonging to the category C, construct
the direct product of G and H as a group in C.
Given a sequence Q of n groups belonging to the category C, construct
the direct product Q[1] x Q[2] x ... x Q[n] as a group
in the category C.
We define G to be the symmetric group of degree 4 and
H to be the dihedral group of order 8. We then form the
direct product of G and H.
> G := SymmetricGroup(4);
> H := DihedralGroup(3);
> D := DirectProduct(G, H);
> D;
Permutation group D acting on a set of cardinality 7
(1, 2, 3, 4)
(1, 2)
(5, 6, 7)
(5, 6)
> Order(D);
144
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