(a) Left and right actions
Let G be a group with identity element e, and let X be a set. An action of G on X is a mapping from to X with the property that, if g x denotes the image of , thenAn element g of G thus induces a mapping of X into itself defined by , with the `representation property':Since G is a group, every g has an inverse ; hence every mapping has an inverse , so that each is a permutation of X.
Strictly speaking, what has just been defined is a left action. A right action of G on X is defined similarly as a mapping such thatThe mapping defined by then has the `rightrepresentation' property:
The essential difference between left and right actions is of course not whether the elements of G are written on the left or right of those of X: it lies in the difference between (iii) and (iii′). In a left action the product in G operates on by operating first, then operating on the result; in a right action, operates first, then . This distinction will be of importance in Sections 1.3.4.2.2.4 and 1.3.4.2.2.5. In the sequel, we will use left actions unless otherwise stated.
(b) Orbits and isotropy subgroups
Let x be a fixed element of X. Two fundamental entities are associated to x:
(1) the subset of G consisting of all g such that is a subgroup of G, called the isotropy subgroup of x and denoted ;
(2) the subset of X consisting of all elements gx with g running through G is called the orbit of x under G and is denoted Gx.

Through these definitions, the action of G on X can be related to the internal structure of G, as follows. Let denote the collection of distinct left cosets of in G, i.e. of distinct subsets of G of the form . Let and denote the numbers of elements in the corresponding sets. The number of distinct cosets of in G is also denoted and is called the index of in G; by Lagrange's theoremNow if and are in the same coset of , then with , and hence ; the converse is obviously true. Therefore, the mapping from cosets to orbit elementsestablishes a onetoone correspondence between the distinct left cosets of in G and the elements of the orbit of x under G. It follows that the number of distinct elements in the orbit of x is equal to the index of in G:and that the elements of the orbit of x may be listed without repetition in the form
Similar definitions may be given for a right action of G on X. The set of distinct right cosets in G, denoted , is then in onetoone correspondence with the distinct elements in the orbit xG of x.
(c) Fundamental domain and orbit decomposition
The group properties of G imply that two orbits under G are either disjoint or equal. The set X may thus be written as the disjoint unionwhere the are elements of distinct orbits and I is an indexing set labelling them. The subset is said to constitute a fundamental domain (mathematical terminology) or an asymmetric unit (crystallographic terminology) for the action of G on X: it contains one representative of each distinct orbit. Clearly, other fundamental domains may be obtained by choosing different representatives for these orbits.
If X is finite and if f is an arbitrary complexvalued function over X, the `integral' of f over X may be written as a sum of integrals over the distinct orbits, yielding the orbit decomposition formula:In particular, taking for all x and denoting by the number of elements of X:
(d) Conjugation, normal subgroups, semidirect products
A group G acts on itself by conjugation, i.e. by associating to the mapping defined byIndeed, and . In particular, operates on the set of subgroups of G, two subgroups H and K being called conjugate if for some ; for example, it is easily checked that . The orbits under this action are the conjugacy classes of subgroups of G, and the isotropy subgroup of H under this action is called the normalizer of H in G.
If is a oneelement orbit, H is called a selfconjugate or normal subgroup of G; the cosets of H in G then form a group called the factor group of G by H.
Let G and H be two groups, and suppose that G acts on H by automorphisms of H, i.e. in such a way that
Then the symbols [g, h] with , form a group K under the product rule:{associativity checks; [] is the identity; has inverse }. The group K is called the semidirect product of H by G, denoted .
The elements form a subgroup of K isomorphic to G, the elements form a normal subgroup of K isomorphic to H, and the action of G on H may be represented as an action by conjugation in the sense that
A familiar example of semidirect product is provided by the group of Euclidean motions (Section 1.3.4.2.2.1). An element S of may be written with , the orthogonal group, and , the translation group, and the product lawshows that with acting on by rotating the translation vectors.
(e) Associated actions in function spaces
For every left action of G in X, there is an associated left action of G on the space of complexvalued functions over X, defined by `change of variable' (Section 1.3.2.3.9.5):Indeed for any in G,since , it follows thatIt is clear that the change of variable must involve the action of (not g) if is to define a left action; using g instead would yield a right action.
The linear representation operators on provide the most natural instrument for stating and exploiting symmetry properties which a function may possess with respect to the action of G. Thus a function will be called Ginvariant if for all and all . The value then depends on x only through its orbit G x, and f is uniquely defined once it is specified on a fundamental domain ; its integral over X is then a weighted sum of its values in D:
The Ginvariance of f may be written:Thus f is invariant under each , which obviously implies that f is invariant under the linear operator in which averages an arbitrary function by the action of G. Conversely, if , thenso that the two statements of the Ginvariance of f are equivalent. The identityis easily proved by observing that the map ( being any element of G) is a onetoone map from G into itself, so thatas these sums differ only by the order of the terms. The same identity implies that is a projector:and hence that its eigenvalues are either 0 or 1. In summary, we may say that the invariance of f under G is equivalent to f being an eigenfunction of the associated projector for eigenvalue 1.
(f) Orbit exchange
One final result about group actions which will be used repeatedly later is concerned with the case when X has the structure of a Cartesian product:and when G acts diagonally on X, i.e. acts on each separately:Then complete sets (but not usually minimal sets) of representatives of the distinct orbits for the action of G in X may be obtained in the formfor each , i.e. by taking a fundamental domain in and all the elements in with . The action of G on each does indeed generate the whole of X: given an arbitrary element of X, there is an index such that and a coset of in G such that for any representative γ of that coset; thenwhich is of the form with .
The various are related in a simple manner by `transposition' or `orbit exchange' (the latter name is due to J. W. Cooley). For instance, may be obtained from as follows: for each there exists and such that ; thereforesince the fundamental domain of is thus expanded to the whole of , while is reduced to its fundamental domain. In other words: orbits are simultaneously collapsed in the jth factor and expanded in the kth.
When G operates without fixed points in each (i.e. for all ), then each is a fundamental domain for the action of G in X. The existence of fixed points in some or all of the complicates the situation in that for each k and each such that the action of on the other factors must be examined. Shenefelt (1988) has made a systematic study of orbit exchange for space group P622 and its subgroups.
Orbit exchange will be encountered, in a great diversity of forms, as the basic mechanism by which intermediate results may be rearranged between the successive stages of the computation of crystallographic Fourier transforms (Section 1.3.4.3).