# 21 Permutation Groups

A permutation group is a group of permutations on a set Omega of positive integers (see chapters Groups and Permutations).

Our standard example in this chapter will be the symmetric group of degree 4, which is defined by the following GAP statements.

gap> s4 := Group( (1,2), (1,2,3,4) );
Group( (1,2), (1,2,3,4) )

This introduction is followed by a section that describes the function that tests whether an object is a permutation group or not (see section IsPermGroup). The next sections describe the functions that are related to the set of points moved by a permutation group (see PermGroupOps.MovedPoints, PermGroupOps.SmallestMovedPoint, PermGroupOps.LargestMovedPoint, and PermGroupOps.NrMovedPoints). The following section describes the concept of stabilizer chains, which are used by most functions for permutation groups (see Stabilizer Chains). The following sections describe the functions that compute or change a stabilizer chain (see StabChain, ExtendStabChain, ReduceStabChain, MakeStabChainStrongGenerators). The next sections describe the functions that extract information from stabilizer chains (see Base for Permutation Groups, ListStabChain, PermGroupOps.Indices, and PermGroupOps.StrongGenerators). The next two sections describe the functions that find elements or subgroups of a permutation group with a property (see PermGroupOps.ElementProperty and PermGroupOps.SubgroupProperty).

If the permutation groups become bigger, computations become slower. In many cases it is preferable then, to use random methods for computation. This is explained in section Random Methods for Permutation Groups.

Because each permutation group is a domain all set theoretic functions Set Functions for Permutation Groups). Also because each permutation group is after all a group all group functions can be applied to it (see chapter Groups and Group Functions for Permutation Groups). Finally each permutation group operates naturally on the positive integers, so all operations functions can be applied (see chapter Operations of Groups and Operations of Permutation Groups). The last section in this chapter Permutation Group Records).

The external functions are in the file LIBNAME/"permgrp.g".

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GAP 3.4.4
April 1997