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.
Permutation groups in Magma are limited to degree less than 230.
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: