One of the most important tools in group theory is the operation or action of a group on a certain set.
We say that a group G operates on a set D if we have a function that takes each d in D and each g in G to another element d^g in D, which we call the image of d under g, such that d^{identity} = d and (d^g)^h = d^{gh} for each d in D and g,h in G.
This is equivalent to saying that an operation is a homomorphism of the group G into the full symmetric group on D. We usually call D the domain of the operation and its elements points.
An example of the usage of the functions in this package can be found in the introduction to GAP (see About Operations of Groups).
In GAP group elements usually operate through the power operator,
which is denoted by the caret ^. It is possible however to specify
other operations (see Other Operations).
First this chapter describes the functions that take a single element of
the group and compute cycles of this group element and related
information (see Cycle, CycleLength, Cycles, and CycleLengths),
and the function that describes how a group element operates by a
permutation that operates the same way on [1..n] (see Permutation).
Next come the functions that test whether an orbit has minimal or maximal length and related functions (see IsFixpoint, IsFixpointFree, DegreeOperation, IsTransitive, and Transitivity).
Next this chapter describes the functions that take a group and compute orbits of this group and related information (see Orbit, OrbitLength, Orbits, and OrbitLengths).
Next are the functions that compute the permutation group P that
operates on [ 1 .. Length(D) ] in the same way that G operates on
D, and the corresponding homomorphism from G to P (see Operation,
OperationHomomorphism).
Next is the functions that compute block systems, i.e., partitions of D such that G operates on the sets of the partition (see Blocks), and the function that tests whether D has such a nontrivial partitioning under the operation of G (see IsPrimitive).
Finally come the functions that relate an orbit of G on D with the subgroup of G that fixes the first point in the orbit (see Stabilizer), and the cosets of this subgroup in G (see RepresentativeOperation and RepresentativesOperation).
All functions described in this chapter are in LIBNAME/"operatio.g".
GAP 3.4.4