Finitely generated groups and their subgroups are important domains in GAP. They are represented as permutation groups, matrix groups, ag groups or even more complicated constructs as for instance automorphism groups, direct products or semi-direct products where the group elements are represented by records.
Groups are created using Group (see Group), they are represented by
records that contain important information about the groups. Subgroups
are created as subgroups of a given group using Subgroup, and are also
represented by records. See More about Groups and Subgroups for
details about the distinction between groups and subgroups.
Because this chapter is very large it is split into several parts. Each part consists of several sections.
Note that some functions will only work if the elements of a group are represented in an unique way. This is not true in finitely presented groups, see Group Functions for Finitely Presented Groups for a list of functions applicable to finitely presented groups.
The first part describes the operations and functions that are available
for group elements, e.g., Order (see Group Elements). The next part
tells your more about the distinction of parent groups and subgroups (see
More about Groups and Subgroups). The next parts describe the
functions that compute subgroups, e.g., SylowSubgroup (Subgroups),
Series of Subgroups). The next part describes the functions that compute and test
properties of groups, e.g., AbelianInvariants and IsSimple (see
Properties and Property Tests), and that identify the isomorphism type.
The next parts describe conjugacy classes of elements and subgroups (see
Conjugacy Classes) and cosets (see Cosets of Subgroups). The next
part describes the functions that create new groups, e.g.,
DirectProduct (see Group Constructions). The next part describes
Group Homomorphisms). The last part tells you more about the implementation
Set Functions for Groups).
The functions described in this chapter are implemented in the following
library files. LIBNAME/"grpelms.g" contains the functions for group
elements, LIBNAME/"group.g" contains the dispatcher and default group
functions, LIBNAME/"grpcoset.g" contains the functions for cosets and
factor groups, LIBNAME/"grphomom.g" implements the group
homomorphisms, and LIBNAME/"grpprods.g" implements the group
constructions.
GAP 3.4.4