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.

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.