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)[link] and Scott (1964)[link] 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 [G \times X] to X with the property that, if g x denotes the image of [(g, x)], then[\displaylines{\quad \hbox{(i)}\,\, (g_{1} g_{2}) x = g_{1} (g_{2}x)\quad \hbox{for all } g_{1}, g_{2} \in G \hbox{ and all } x \in X, \hfill\cr \quad \hbox{(ii)}{\hbox to 22pt{}}ex = x {\hbox to 34pt{}}\hbox{for all } x \in X. \hfill}]An element g of G thus induces a mapping [T_{g}] of X into itself defined by [T_{g} (x) = gx], with the `representation property':[\displaylines{\quad \hbox{(iii) }T_{g_{1} g_{2}} = T_{g_{1}} T_{g_{2}} \quad \hbox{for all } g_{1}, g_{2} \in G.\hfill}]Since G is a group, every g has an inverse [g^{-1}]; hence every mapping [T_{g}] has an inverse [T_{g^{-1}}], so that each [T_{g}] 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 [(g, x) \,\longmapsto\, xg] such that[\displaylines{\quad (\hbox{i}')\,\,\, x(g_{1} g_{2}) = (xg_{1}) g_{2}\quad \,\hbox{ for all } g_{1}, g_{2} \in G \hbox{ and all } x \in X, \hfill\cr \quad (\hbox{ii}'){\hbox to 25pt{}} xe = x{\hbox to 39pt{}}\hbox{for all } x \in X. \hfill}]The mapping [T'_{g}] defined by [T'_{g}(x) = xg] then has the `right-representation' property:[\displaylines{\quad (\hbox{iii}')\ T'_{g_{1} g_{2}} = T'_{g_{2}} T'_{g_{1}}\quad \hbox{for all } g_{1}, g_{2} \in G.\hfill}]

    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 [g_{1} g_{2}] in G operates on [x \in X] by [g_{2}] operating first, then [g_{1}] operating on the result; in a right action, [g_{1}] operates first, then [g_{2}]. This distinction will be of importance in Sections 1.3.4.2.2.4[link] and 1.3.4.2.2.5[link]. 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 [gx = x] is a subgroup of G, called the isotropy subgroup of x and denoted [G_{x}];

    • (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 [G / G_{x}] denote the collection of distinct left cosets of [G_{x}] in G, i.e. of distinct subsets of G of the form [gG_{x}]. Let [|G|, |G_{x}|, |Gx|] and [|G / G_{x}|] denote the numbers of elements in the corresponding sets. The number [|G / G_{x}|] of distinct cosets of [G_{x}] in G is also denoted [[G : G_{x}]] and is called the index of [G_{x}] in G; by Lagrange's theorem[[G : G_{x}] = |G/G_{x}| = {|G| \over |G_{x}|}.]Now if [g_{1}] and [g_{2}] are in the same coset of [G_{x}], then [g_{2} = g_{1}g'] with [g' \in G_{x}], and hence [g_{1}x = g_{2}x]; the converse is obviously true. Therefore, the mapping from cosets to orbit elements[gG_{x} \,\longmapsto\, gx]establishes a one-to-one correspondence between the distinct left cosets of [G_{x}] 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 [G_{x}] in G:[|Gx| = [G : G_{x}] = {|G| \over |G_{x}|},]and that the elements of the orbit of x may be listed without repetition in the form[Gx = \{\gamma x | \gamma \in G/G_{x}\}.]

    Similar definitions may be given for a right action of G on X. The set of distinct right cosets [G_{x}g] in G, denoted [G_{x} \backslash G], 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 union[X = \bigcup\limits_{i \in I} Gx_{i},]where the [x_{i}] are elements of distinct orbits and I is an indexing set labelling them. The subset [D = \{x_{i}\}_{i\in I}] 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 [x_{i}] 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:[\eqalign{{\sum\limits_{x\in X}}\, f(x) &= {\sum\limits_{i\in I}} \left({\sum\limits_{y_{i}\in Gx_{i}}} f(y_{i})\right) = {\sum\limits_{i\in I}} \left({\sum\limits_{\gamma_{i}\in G/G_{x_{i}}}} f(\gamma_{i} x_{i})\right) \cr &= \sum\limits_{i\in I} {1 \over |G_{x_{i}}|} \left(\sum\limits_{g_{i}\in G} f(g_{i} x_{i})\right).}]In particular, taking [f(x) = 1] for all x and denoting by [|X|] the number of elements of X:[|X| = \sum\limits_{i\in I} |Gx_{i}| = \sum\limits_{i\in I} |G/G_{x_{i}}| = \sum\limits_{i\in I} {|G| \over |G_{x_{i}}|}.]

  • (d) Conjugation, normal subgroups, semi-direct products

    A group G acts on itself by conjugation, i.e. by associating to [g \in G] the mapping [C_{g}] defined by[C_{g} (h) = ghg^{-1}.]Indeed, [C_{g} (hk) = C_{g} (h) C_{g} (k)] and [[C_{g} (h)]^{-1} = C_{g^{-1}}(h)]. In particular, [C_{g}] operates on the set of subgroups of G, two subgroups H and K being called conjugate if [H = C_{g} (K)] for some [g \in G]; for example, it is easily checked that [G_{gx} = C_{g}(G_{x})]. 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 [\{H\}] 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 [G/H] 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[\eqalign{g (h_{1}h_{2}) &= g(h_{1})g(h_{2}) \cr g (e_{H}) &= e_{H}\qquad\quad (\hbox{where } e_{H} \hbox{ is the identity element of } H). \cr g (h^{-1}) &= (g(h))^{-1}}]

    Then the symbols [g, h] with [g \in G], [h \in H] form a group K under the product rule:[[g_{1}, h_{1}] [g_{2}, h_{2}] = [g_{1}g_{2}, h_{1}g_{1}(h_{2})]]{associativity checks; [[e_{G},e_{H}]] is the identity; [[g,h]] has inverse [[g^{-1}, g^{-1} (h^{-1})]]}. The group K is called the semi-direct product of H by G, denoted [K = H\, \triangleright\kern-4pt \lt G].

    The elements [[g, e_{H}]] form a subgroup of K isomorphic to G, the elements [[e_{G}, h]] 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[C_{[g, \, e_{H}]} ([e_{G}, h]) = [e_{G}, g(h)].]

    A familiar example of semi-direct product is provided by the group of Euclidean motions [M(3)] (Section 1.3.4.2.2.1[link]). An element S of [M(3)] may be written [S = [R, t]] with [R \in O(3)], the orthogonal group, and [t \in T(3)], the translation group, and the product law[[R_{1}, t_{1}] [R_{2}, t_{2}] = [R_{1}R_{2}, t_{1} + R_{1}(t_{2})]]shows that [ M(3) = T(3) \, \triangleright\kern-4pt \lt O(3)] with [O(3)] acting on [T(3)] by rotating the translation vectors.

  • (e) Associated actions in function spaces

    For every left action [T_{g}] of G in X, there is an associated left action [T_{g}^{\#}] of G on the space [L(X)] of complex-valued functions over X, defined by `change of variable' (Section 1.3.2.3.9.5[link]):[[T_{g}^{\#} f](x) = f((T_{g})^{-1} x) = f(g^{-1} x).]Indeed for any [g_{1},g_{2}] in G,[\eqalign{[T_{g_{1}}^{\#}[T_{g_{2}}^{\#}\, f]](x) &= [T_{g_{2}}^{\#}\, f] ((T_{g_{1}})^{-1} x) = f[T_{g_{2}}^{-1} T_{g_{1}}^{-1} x] \cr &= f((T_{g_{1}} T_{g_{2}})^{-1} x)\hbox{\semi}}]since [T_{g_{1}} T_{g_{2}} = T_{g_{1}g_{2}}], it follows that[T_{g_{1}}^{\#} T_{g_{2}}^{\#} = T_{g_{1}g_{2}}^{\#}.]It is clear that the change of variable must involve the action of [g^{-1}] (not g) if [T^{\#}] is to define a left action; using g instead would yield a right action.

    The linear representation operators [T_{g}^{\#}] on [L(X)] 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 [f \in L(X)] will be called G-invariant if [f(gx) = f(x)] for all [g \in G] and all [x \in X]. The value [f(x)] then depends on x only through its orbit G x, and f is uniquely defined once it is specified on a fundamental domain [D = \{x_{i}\}_{i\in I}]; its integral over X is then a weighted sum of its values in D:[{\textstyle\sum\limits_{x \in X}}\, f(x) = {\textstyle\sum\limits_{i \in I}}\, [G:G_{x_{i}}]\, f(x_{i}).]

    The G-invariance of f may be written:[T_{g}^{\#}f = f \quad \hbox{for all } g \in G.]Thus f is invariant under each [T_{g}^{\#}], which obviously implies that f is invariant under the linear operator in [L(X)][A_{G} = {1 \over |G|} \sum\limits_{g \in G} T_{g}^{\#},]which averages an arbitrary function by the action of G. Conversely, if [A_{G}\,f = f], then[T_{g_{0}}^{\#}\, f = T_{g_{0}}^{\#} (A_{G}\,f) = (T_{g_{0}}^{\#} A_{G})f = A_{G}\,f = f \quad \hbox{for all } g_{0} \in G,]so that the two statements of the G-invariance of f are equivalent. The identity[T_{g_{0}}^{\#} A_{G} = A_{G} \quad \hbox{for all } g_{0} \in G]is easily proved by observing that the map [g \,\longmapsto\, g_{0}g] ([g_{0}] being any element of G) is a one-to-one map from G into itself, so that[{\textstyle\sum\limits_{g \in G}} T_{g}^{\#} = {\textstyle\sum\limits_{g \in G}} T_{g_{0}g}^{\#}]as these sums differ only by the order of the terms. The same identity implies that [A_{G}] is a projector:[(A_{G})^{2} = A_{G},]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 [A_{G}] 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:[X = X_{1} \times X_{2} \times \ldots \times X_{n}]and when G acts diagonally on X, i.e. acts on each [X_{j}] separately:[gx = g(x_{1}, x_{2}, \ldots, x_{n}) = (gx_{1}, gx_{2}, \ldots, gx_{n}).]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 form[D_{k} = X_{1} \times \ldots \times X_{k-1} \times \{x_{i_{k}}^{(k)}\}_{i_{k} \in I_{k}} \times X_{k+1} \times \ldots \times X_{n}]for each [k = 1, 2, \ldots, n], i.e. by taking a fundamental domain in [X_{k}] and all the elements in [X_{j}] with [j \neq k]. The action of G on each [D_{k}] does indeed generate the whole of X: given an arbitrary element [y = (y_{1}, y_{2}, \ldots, y_{n})] of X, there is an index [i_{k} \in I_{k}] such that [y_{k} \in Gx_{i_{k}}^{(k)}] and a coset of [G_{x_{i_{k}}^{(k)}}] in G such that [y_{k} = \gamma x_{i_{k}}^{(k)}] for any representative γ of that coset; then[y = \gamma (\gamma^{-1} y_{1}, \ldots, \gamma^{-1} y_{k-1}, x_{i_{k}}^{(k)}, \gamma^{-1} y_{k+1}, \ldots, \gamma^{-1} y_{n})]which is of the form [y = \gamma d_{k}] with [d_{k} \in D_{k}].

    The various [D_{k}] are related in a simple manner by `transposition' or `orbit exchange' (the latter name is due to J. W. Cooley). For instance, [D_{j}] may be obtained from [D_{k}(\,j \neq k)] as follows: for each [y_{j} \in X_{j}] there exists [g(y_{j}) \in G] and [i_{j}(y_{j}) \in I_{j}] such that [y_{j} = g(y_{j})x_{i_{j}(y_{j})}^{(j)}]; therefore[D_{j} = \bigcup\limits_{y_{j} \in X_{j}} [g(y_{j})]^{-1} D_{k},]since the fundamental domain of [X_{k}] is thus expanded to the whole of [X_{k}], while [X_{j}] 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 [X_{k}] (i.e. [G_{x_{k}} = \{e\}] for all [x_{k} \in X_{k}]), then each [D_{k}] is a fundamental domain for the action of G in X. The existence of fixed points in some or all of the [X_{k}] complicates the situation in that for each k and each [x_{k} \in X_{k}] such that [G_{x_{k}} \neq \{e\}] the action of [G/G_{x_{k}}] on the other factors must be examined. Shenefelt (1988)[link] 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)[link].

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.








































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