A permutation group G is a group of bijections X to X, for some set X. The group G is said to act on X and the elements of G are called permutations (of the set X). A given permutation group G may have actions on sets other than the one on which it is defined. Thus, any set upon which G has a legitimate action will be called a G-set. The set X is called the natural G-set for the group G, and the action of G on X is called the natural action of G. Note that the group G also has a natural induced action on the G-closure of any derived set of X . Magma expects the G-set X to be of finite cardinality n. Usually, X will be {1, 2, ..., n}, but, as we shall see below, X may be a set of strings, or any other legitimate Magma set.
The elements of a G-set are called points. Let Y be a G-set for G. The (possibly empty) subset of Y whose points are fixed by every permutation of G, is called the fixed-point set for G, while the subset of Y consisting of points moved by some permutation of G is called the support of G. Similarly, for an element g of G the fixed-point set and the support of g are, respectively, the subsets of Y consisting of the points fixed and moved by g. The degree of G is defined to be the cardinality of the natural G-set of G; whereas the degree of an element g of G is defined to be the cardinality of the support of g, i.e. the number of points moved by g.
The family of all permutation groups of finite degree forms a category. The objects are the permutation groups and the morphisms are group homomorphisms. The Magma designation for this category of permutation groups is GrpPerm.
Every permutation group acting on a set X is created as a subgroup of the symmetric group Sym(X). Thus, the construction of a general permutation group is a two-step process: