Groups

Magma supports the following categories of groups:

GrpAb

abelian groups

GrpPerm

permutation groups

GrpFP

finitely presented groups

GrpPC

soluble groups in polycyclic (power-conjugate) presentations

GrpMat

matrix groups

GrpBB

blackbox groups

The construction of these groups is category-dependent, and may be performed in several ways. See the relevant help pages for details; the following notes are intended as an overview.

The generic `full' or `free' magmas are created by the functions

FreeAbelianGroup

for GrpAb

SymmetricGroup or Sym

for GrpPerm

FreeGroup

for GrpFP and GrpPC

GeneralLinearGroup or GL

for GrpMat

BlackboxGroup

for GrpBB

One may create any group by creating one of these structures and then taking the required subgroup (for GrpPerm and GrpMat) or quotient group (for the structures with presentations). The sub or quo constructor should be used for this purpose.

To abbreviate this process, the following constructors apply:

• AbelianGroup
• PermutationGroup
• Group
• PolycyclicGroup
• MatrixGroup

There are special functions for constructing some standard groups. These functions take the required category of group (such as GrpFP) as their first argument, but are generally not available for all categories:

• CyclicGroup
• DihedralGroup
• AbelianGroup
• AlternatingGroup
• SymmetricGroup

If the category argument is omitted then a permutation group will be constructed (except for AbelianGroup). The GrpMat category has a separate collection of standard groups, namely GL, SL, GU, SU, Sp, and Sz.

The order of group G is Order(G) or #G. If the order cannot be computed then the value 0 is returned.

The operators for arithmetic on group elements are:

• g * h
• g ^ n (including g ^ -1 for the inverse)
• g ^ h
• (g, h) for the commutator (NOT [g, h], which is a sequence).

The order of element g is Order(g). The identity is Id(G) or G!1.

Functions testing properties of a group include

• IsSoluble/IsSolvable

• IsNormal
• IsConjugate
• IsMaximal

These functions and operators return some standard subgroups:

• SylowSubgroup
• H meet K
• H ^ g
• Core
• Centralizer
• Normalizer
• ncl< G | H >

In order to transfer from one category of group to another, the following functions exist:

• FPGroup
• PCGroup
• CosetAction
• PermRepLimit

See the help nodes on these topics for further information.

Example

> D<s, t> := DihedralGroup(GrpPC, 7); 
> print D;
GrpPC : D of order 14 = 2 * 7
PC-Relations:
s^2 = Id(D), 
t^7 = Id(D), 
t^s = t^6
> DH := sub< D | t^3 >;
> print DH;
GrpPC : DH of order 7
PC-Relations:
DH.1^7 = Id(DH)
> print IsCyclic(DH);
true