Introduction

Groups arise in several different categories in Magma. In the case of the category of permutation groups and the category of soluble groups defined by a power-conjugate presentation, all groups in the category are finite. However, the finitely-presented group category, the polycyclic group category, the abelian group category and the matrix group category contain both finite and infinite groups. In the case of the abelian group category and the matrix group category, a large number of functions are available for finite groups only. In the near future, these functions will be extended to finite finitely-presented groups of moderate order.

In this chapter, we discuss the functions that are provided for groups collectively, noting especially those functions that are available only for finite groups. Descriptions of functions that depend upon the particular category may be found in the chapter devoted to that category.

Contents

The Categories of Finite Groups

At present Magma contains five main categories of finite groups:

(i)
Permutation groups: category GrpPerm;
(ii)
Finite matrix groups: category GrpMat;
(iii)
Finite solvable groups given by a power-conjugate presentation: category GrpPC;
(iv)
Finite abelian groups: category GrpAb;
(v)
Finite polycyclic groups: category GrpGPC.

Note that the categories GrpMat, GrpAb and GrpGPC contain both finite and infinite groups; most of the operations described in this chapter apply only to finite groups belonging to these categories. In this chapter we will use the category name GrpFin to collectively refer to categories GrpPerm and GrpPC and the subcategories of GrpMat, GrpAb and GrpGPC consisting of finite groups. The category name Grp will be used when the operation does not depend upon the finiteness of the group.

V2.28, 13 July 2023