International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 68-70   | 1 | 2 |

## Section 1.3.4.2.2.2. Groups and group actions

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.4.2.2.2. Groups and group actions

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The books by Hall (1959) and Scott (1964) are recommended as reference works on group theory.

• (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 right-representation' 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 one-to-one 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 one-to-one 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 complex-valued 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, semi-direct 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 one-element orbit, H is called a self-conjugate 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 semi-direct 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 semi-direct 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 complex-valued 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 G-invariant 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 G-invariance 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 G-invariance of f are equivalent. The identityis easily proved by observing that the map ( being any element of G) is a one-to-one 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).

### References

Hall, M. (1959). The Theory of Groups. New York: Macmillan.
Scott, W. R. (1964). Group Theory. Englewood Cliffs: Prentice-Hall. [Reprinted by Dover, New York, 1987.]
Shenefelt, M. (1988). Group Invariant Finite Fourier Transforms. PhD thesis, Graduate Centre of the City University of New York.