E. 1. Burrows and M. J. Clark University of E O S ~Anglia University Plain, Norwich, NOR 88C. England
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Pictures of Point Groups
At an introductory level in group theory, questions often arise such as, "What is the difference between a group element and a symmetry element?" or "Why do different point groups have the same character table (e.g., Cs,, D3)?" These questions cannot always he answered satisfactorily because they arise from the abstract nature of group theory. Physical scientists, and chemists in particular, are taught to think,in three dimensions so that they appreciate the structure of physical models. In an attempt to reconcile the ahstract and the physical this article deals with the pictorial representation of the abstract structure of groups. In our experience, as well as being useful in practical applications of group theory, these pictures are a valuable teaching aid.1 We begin by showing the corres~ondencesbetween mouDs and their ~ictorial representa&ons (or graphs) closely following ~ a b u and s Grossman's excellent book ( I ) . These graphs are then related to groups of symmetry operations and we include a comprehensive collection of pictorial representations of important point groups. Cayley Tables and Diagrams By definition a group is a set of elements with a rule of comhination satisfying certain postulates. These postulates are identity, closure, inverse, and associativity, and in order to completely specify a group, it is only necessary to show how the various elements combine. This is normally achieved with a Cayley (multiplication) table. For example, the symmetry group for the molecule N(Ph)a, Figure 1 (a), is the Ca point group which is specified by the tahle shown in Figure 1(h). This method of displaying fundamental group properties is well known and was pioneered by Cayley (2). However, Cayley also developed a pictorial method of representing the way in which group elements combine (3)and consequently these pictures are called Cayley diagrams (4). T o illustrate the method we can consider the ahstract properties of the point group C 3 . The symmetry operation C3 can be represented algebraically by the symbol a, and the ahstract Cayley table is shown in Figure 2 (a) with the Cayley diagram or graph of the group in Figure 2 (h). The graph acts as a compact display of the multiplication tahle because each point corresponds to a distinct group element, and proceeding round the graph in the direction of the arrows corresponds to multiplication by a. In going from I to a we have a pictorial representation of the group multiplication I X a = a, and going from a to a2 corresponds to the multiplication o X a = a2. In addition to showing how the group elements combine, the graph embodies the axioms of group theory. 1) There is a unique identity element I (the choice of where we
put I on the graph is arbitrary, since it is a symmetric graph). 2) The graph is a closed figure so comhination of any two elernentsgwes an element wthin the proup 3 ) I n \ e r w can he reulmd bv proceeding agamst the dinctinn of the arrows. tl'hp direered line rorresvundi t u multmliration by a so if we go from I against the direction of the &ow we arrive at 02, which is the inverse of a). We have not included the possibility of non-commutation of group elements (ab f ba) and so there is a con-
Figure 1. (a) Triphenylamine and (b) the corresponding Cayley table.
Figure 2. (a) Cayley table and (b) diagram
Figure 3. The congruence rotations (in its own plane) of a triangle
vention regarding the direction of the line and right hand multiplication.2 When we proceed fmm a2 to I we mean multiply a2 by a on the right, a2 x a not a x a=. Our present example of a group of elements, ll,a,a21 is obviously commutative (ahelian). We shall introduce an example of a non-cummutative group (non-abelian group) below. Generators and Defining Relations The elements of the group Ca {I,a,a2) are expressed in terms of powers of a with the exception of the unique element I. However. I is also exnressihle in terms of a as shown by considering the congruence rotations of a triangle. Figure 3 illustrates these rotations of a triangle in its ' A video tape devised by the authors and called "The Group Postulates" has been used to introduce first year chemistry undergraduates at U.E.A. to the subject of group theory. This is the convention adopted by mathematicians and is a consequence of their definition of group multiplication. For erample, Magnus and Grossman (n use the term "Followed by" to introduce the idea of multiplication, and define it with respect to the graphs. "Moving in the direction of the arrows corresponds to right multiplication by the element a,"
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own plane (one comer of the triangle has been labelled arbitrarily). If we also label one configuration I, a rotation by 120" takes us to configuration a, a further rotation takes us to a2, and then such a rotation takes the triangle to its original configuration I. We could do the same seGrn& : o quence oioperationi with the triphenylamine molecule by Rebtlon 02 = I arbitrarily labeling one of the phenyl groups. The imporPant Gmups : CI lo=GI tant result is that the rotation (or the Ca o~eration)gent, I0 = 01 erates all the group elements, and therefore is c a ~ l e ~ t h e C ibil generator of the group. Also we have discovered the equation a3 = I (or C3J = 0 which is called the defining relation of the group. All groups are expressable in terms of generators and defining relations (5), and mathematicians often prefer to do so because such a specification is independent of the order of a group. For example, it is easier to specify the infinite rotation group by the generator a and defining relation a" = I where n than to attempt to write out an infinite multiplication table ( 0 . Similarly the point group O h can he presented in terms of two generators (6) and three defining relations which contain all the information about the combination of the 48 group elements. So in generators and relations we have the minimum amount of information that is required to define a group, and the graphs provide a pictorial representation of this information. Formally we can establish the following The graphs of the cyclic groups are relatively straightcorrespondences between groups and graphs ( 1 ) forward due to the need for only one generator, and in Figure 5 we illustrate the more important groups along 1) Each point in a graph corresponds to one gmup element. with the correspondence of the abstract graph to the re2) Proceeding from one point to another along a directed line spective point groups. We note that the point groups S4 (in the direction of an arrow) corresponds to multiplication and $6 are cyclic groups where the generator is an im(on the R.H.S.)by a generator of the group. The distance proper rotation. traversed is called a path. If we proceed against the direction of the arrow it corresponds to multiplication by the inGroups with Two Generators verse of the generator. Earlier we showed how the group elements of the point 3) Lines of different type represent multiplication by different group C3 combine and used N(Ph)3 as an example of the generators. (We have used full lines and dotted Lines to show differentgenerators). group. The group elements {I,C3,C32) formed a cyclic 4) A succession of paths corresponds to multiplication of elegroup which could be related to the abstract structure ments. {I,a,a2).Suppose we introduce another generator, b, such that bz = I. Lwking at the N(Ph)3 example again, we Finally the graph of a could replace each twisted phenyl group by an isotropic group must be a symmetrihydrogen atom, therefore introducing reflection as a new cally connected graph with symmetry operation. This could abstractedly be reprethe same number of lines a t sented by 0 2 = b2 = I. The new symmetry group is the C3" each point, and each point point group which has the multiplication table shown in Figure 6. I must be reachable along I some path from every other The abstract structure of the group is shown in Figure 7 i I point. If we consider the 6 graph in Figure 4 which is I I I not symmetrical, we can quickly see that one of the of gmup theory is violated, namely that every element must have a unique Figure 4. Graph not related to a inverse. In the graph there is group strucfure. no inverse for the group element b.
a
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Groups with One Generator These groups are normally called the cyclic groups and an already familiar example will show why this nomenclature is suitable. If we consider again our group of the congruence rotations of a triangle, Figure 3 taking higher and higher powers of the generator a gives the series a, n2. 03, a4, a" us, a', an,as, a10
Figure 6. Multiplication table for 6 3 0
. ..
but if we substitute the defining relation a3 = I we obtain the series a,a2,1,a,a2,1,a,a?I,o...
So we have a cyclic pattern of the elements a, a2,I. If we had a different defining relation (say a5 = I, for the congruence rotations of a pentagon) we would get a different cyclic pattern. 88 / '~ournalof Chemical Education
Figure 7 . Graph of Csu
with two generators a(Ca) and b ( u P with the defining relations a3 = bZ = abab = I. The graph quickly shows that Csu is a non-abelian group (ab f ba) since ab and ba represent different points on the graph, and correspond to the symmetry operations ~ " ( 3 'and in the point group C3". The graph is symmetrical with an equal number of lines a t each point, and we could put I a t any point as long as the relations a3 = bZ = abab = Iwere retained. Suppose we generate the group D3 by equating a = Cs and b = Cz'. We have again the defining relations a3 = bZ = abab = I, so we obtain the same graph as for C3" (Fig. 8). Therefore, although D3 and C3" are different point groups they have the same abstract c, structure and therefore the same I graph. It is this isomorphism Figure 8. Graph of D, (same structure) which is re-
Figure 10. The cubic groups with two generators.
sponsihle for the two groups having the same character tahle. In group theory terminology they are isomorphic groups, with a 1:l correspondence of group elements. Figures 9 and 10 illustrate the graphs of the point groups with two generators. These include a large proportion of the 32 crystallographic point groups. The table includes the Oh point group which is normally represented by 3 generators (6). Perhaps the most interesting graphs are those of the point groups T, Th, 0 and Td, the socalled "cubic point groups" shown in Figure 10. Their graphs correspond to regular polyhedra. T is represented by a truncated tetrahedron, Th by a truncated cube, and O and Td are represented by the (small) rhomhicuboctahedron. (For our purposes we have projected these polyhedra onto one face.) Groups with Three Generators The groups requiring three generators for specification are Dzn, D4h. Dm and D-h, and we have illustrated their graphs in Figure 11. These are perhaps more complicated than groups with one or two generators (with the notable exception of Oh) hecause more defining relations are required to specify the group. For example, D4h is specified hv the relations a4 = bZ = cZ = ( ~ b =) (bcIZ ~ = ( a ~=) I,~ whereas groups with one generator require one defining relation and those with two generators need three defining relations.
tneraiors . m, b Relul#ms 04 = b2= lob]"
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Oh la = C,
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b
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Figure 9. The dihedral groups with twogeneratars.
Further Correspondences Because the graphs are pictorial representations of the abstract group structure, their main application is in presenting various concepts that are of use in group theory. We have already shown how the graphs embody the axioms of group theory and the "symmetry" of the graphs helps one visualize a group when he has no knowledge of symmetry operations or axes. This helps to minimize any confusion over the concepts of "symmetry operation" and "symmetry element." The graphs show that the group elements used in chemistry are the symmetq operations. Volume 51, Number 2. February 1974
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group then M and N are conjugate. (This equation is called a similarity transform because it shows the effect of a change of coordinate system in operators-certain operators behave similarlv (8).For m o u- ~ sof a reasonable size this calculation is tedious, especially if matrix representations are used. The tedium can he considerablv reduced by using the graphs and the corresponding transform expressed in path lengths. (path l)(poth 2)(path I)-' = (path conjugate to path 2) For example, in the graph in Figure 13 which corresponds to the point groups D, and Cr,, (a)(b)(a)-I = (ba2) using the path conjugation equation: starting at I, ~roceedalong the solid line to a. then along tge dotted lines to ab, and finally along the solid line against the direction of the arrow to ba2. Thus b and ba2 are conjugate and i t is easy to calculate all the classes, [I),{a21,{a, a3), (b, ba2), Figure 13. Abstract graph tor groups Dn and C,,. (ba, abl in this way. Finally let us consider subgroups. Because subgroups are defined as a group within larger groups they can be displayed as graphs within larger graphs. Thus the sets of elements {I, a, a2, a3) and 11, bJ are easily identified as subgroups from Figure 13, and we can say that the graph indicates that C1, has subgroups C4 and C,, and Da has subgroups Ca and Cz. However there are subgroup sets (I, a2, b, ba2) and (I, a21 corresponding to C2" and CZ for CI". and D Z and Cz for D4, which are not so immediately obvious from the graph. (They are, however, still graphs within graphs.) As the complexity of the graphs increases it becomes more difficult to see the suhgroup structure because the graph is dependent on the choice of generators. For example, inspection of the graph for T i n Figure 10 shows that I, a, a2 form a suhgroup corresponding to C3, hut it is not an outstanding feature of the graph that {I,b, azba, abazl is also a suhgroup which corresponds to Dz. Nevertheless for the simpler groups, a group within a group is readily detected as a graph within a graph. v
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Figure 11. The dihedral groups with
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three generators.
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Figure 12. Graphsof groups (a) Cz and (bJ 0 2 . The identity element is the operation of doing nothing, and Cz(z) is the operation of rotation of about 180" along the z axis-neither are symmetry element^.^ As another application consider the symmetrical structure of the graphs which shows why "the existence of two Cz operations along different axes necessarily implies the existence of a third Cz operation along another different axis." The graph of the point group Cz is shown in Figure 12 (a) and the defining relation for the group is C&) = I. If we add a second generator Cz(x) with the defining relation Cz2(x) = I, we obtain the graph in Figure 12 (b). Because our graph must he symmetric (same number and direction of lines a t each point), contained in the graph is a new defining relation ( C z ( z ) . C ~ ( x ) )=~ I where Cz(z).Cz(x) = Czb). Algebraically we have two generators a and b with defining relations a Z = b2 = I and it is impossible to construct a "symmetrical" structure without defining some extra relation such as (ab)z = I. Because this abstract structure is common to the point groups 02, Czu and Czh we can also say that "the existence of a Cz(z) operation and a a, operation necessarily implies the existence of another a,' operation" or "the existence of a Cz(z) operation and an inversion operation necessarily implies the existence of ah operation." As further application we can consider classes which are defined as a set of conjugate (or equivalent) group elements. The class structure is fundamental to most applications of group theory because the number of classes is equal to the number of irreducible representations (7). In order to divide a group into classes we must calculate the sets of equivalent group elements using the similarity transform, which is defined by the equation M = XNX-I where M, N and X are arbitrary elements. If two elements, M and N, satisfy this transform for all X's in the 90
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Summary and Conclusions We have introduced the graphs of point groups and shown how they may he used to illustrate fundamental definitions and properties of groups. Also some applications have heen presented hut we refer the interested reader to Magnus and Grossman ( I ) and a very recent book by Anderson (9) for the important topics of homomorphism, factor groups, invariant subgroups, and permutation groups. The application of graphical representations to research problems in the physical sciences is just beginning (lo), hut considerable theoretical work is required on the connection between generator theory and matrix representation theory before fundamental new insights are obtained. We would like to thank Professor S. F. A. Kettle for his stimulating interest and encouragement, and we would also like to thank the S.R.C. for partial support. Literature Cited (I) cmssman, I., and Magnus, W., "Gmupr
and Their Graphs." New Mathemsfieel Library. Random Hause and L. W. Sineer Co.. New York. 1966. (2) Cayley.A..Pmr.LandanMafh. Soc., 9. 126118781;Amer. J.Mnlh.. 1,50118781. (31 Cayley.A..AmorJ Moth.. I, 17411878). (4, Msgnw. W., Kanass. A,, and Solitar. D.."Combinational Group Theory," John Wiley & Sans Inc., NswYork. 1966. (5) Mcweeney. R.,"Symmetry. AnIntroduction to GmupThoom and Its Applieafions." TheMacrnillanCo.. NewYork. 1963.p.66.
planes about which the symmetry operations are performed.
(6) Waehtman, J.. and Peiser, H . . J Phys. Chem Solids, 27,915 (1966). (7) Hoehatrauner, b b i n M.. "Molecular Aswets of Symmetry," W. A. Benjamin Inc.. New York. 1966. p. 97. (8) Higman. B.. "Applied G m p Theoretie and Mahix Methods." Dover Publications
hc.. New Ynk. p. S?. (9) Andamn. Sabra 5 , ,'Graph Theory and Finite Cambinatarica," M d h a m Pub1lsbingCo.Chicago. 1970. (10) Killingbock. J., J Phys. iC): Solid StatePhya.. 5.732 (1972).
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