Groups and Group Elements

Groups can be given in various forms: for example as permutation groups or matrix groups (by generating elements), as finitely presented groups or as polycyclicly presented groups. GAP knows how to construct a number of well-known groups such as symmetric and classical groups and to fetch concrete groups from group libraries.

There is a wide variety of functions for the investigation of groups. Some of these functions just build on the concept of a group while others (usually the more efficient ones, for instance nearly linear methods for permutation groups) utilize the way in which a particular group is given. GAP tries automatically to select a good method, but the user can take over full control of this selection of methods. Also, if no deterministic method exists (e. g., for determining the order of an fp-group) GAP will try to find an isomorphism to a group it can handle (in the above case it will try to find an isomorphism to a permutation group using the Todd-Coxeter method).

There are many functions to compute invariants of groups, e. g.:

• order (called ‘size’ in GAP),
• conjugacy classes of elements,
• derived series,
• composition series (including identification of the composition factors),
• Sylow subgroups,
• certain characteristic subgroups,
• maximal subgroups,
• normal subgroups,
• subgroup lattice,
• table of marks,
• automorphism group (see also the package AutPGrp),
• cohomology groups (packages cohomolo and HAP and manual section 1-cohomology),
• ordinary character table (see also the page Representations and Characters of Groups).

There are also functions for

• group homomorphisms,
• group actions,
• group products, and
• group constructions
• construction of groups of cube-free order.

Of course the range of applicability of the particular functions depends very much on the order and structure of the group. To give an idea of capabilities, GAP has (already in 1993) been used to find the composition series, Sylow subgroups and character table of a certain solvable subgroup of order 3,265,173,504 in the sporadic simple group Fi₂₃, given as a permutation group of degree 31,671.