Basic Capabilities

GAP provides many ways of forming …

• lists (mutable and immutable), which also coverssets,row vectors,matrices,strings, andBoolean lists,
• records,
• domains, i. e., objects representing sets with an algebraic structure.

Notions and tools modeled in some analogy to elementary set theory help to handle these:

• distinction ofobjects and elements,
• Booleans,
• orderings,
• mappings (in particular mappings respecting structures which again can be elements of structures),
• relations.

GAP can compute with

• arbitrary integers,
• rational numbers,
• cyclotomic numbers, in particularGaussian numbers,
• elements offinite fields (see also the coding theory package GUAVA),
• p-adic numbers,
• polynomials, includingmultivariate polynomials (GAP contains only a basic implementation of Buchberger’s algorithm for computingGröbner bases, for more serious computations the package singular provides an interface to the extensive capabilities of the system SINGULAR),
• rational functions as well as
• various kinds of group elements, e. g.permutations,matrices, andabstract words.

One can work with many algebraic structures. In addition to those listed under Mathematical Capabilities, there are e. g.

• finite fields,
• residue class rings of integers (see also the package ResClasses),
• abelian number fields,
• lattices and integral matrices.

Also there are

• variouscombinatorial functions,
• functions forelementary number theory as well as
• functions forprime number factorization (see also the package FactInt).