Symmetry Groups

DEFINITION. An object is symmetrical if, afterrotation, reflection, or some other rigid motion,it coincides with itself.DEFINITION. A rigid motion is one that presevesthe distances between any two points ofspace. A rigid motion is also called asymmetry operation or an isometry.isometry = iso + metric = same measure1

DEFINITION. The points of an object that areleft fixed by a symmetry operation constitutea symmetry element of that object.For example, in two dimensions these include:OPERATION NOTATION ELEMENTrotation r pointreflection m line2

Another symmetry operation in the plane istranslation t, a parallel shift. Translations haveno fixed points.3

But we’ll leave translation for another lecture.This morning, we consider only symmetry operationswith at least one fixed point. In 3Dthese includeOPERATION NOTATION ELEMENTinversion i pointrotation r linereflection m plane4

We can illustrate the effect of a symmetry operationon an object by marking any point onthe object and its image under that action.If two symmetry operations have the same effecton all points, they are considered to bethe same.For example, for an object IN THE PLANE(but not in 3D), 180deg rotation and inversionin the center are the same.5

One symmetry operation followed by anotheris another symmetry operation. ”Followed by”is a kind of multiplication or product ◦. Forthe square below,reflection in m1 ◦ reflection in m2 = rwhere r is 45deg rotation about the center ofthe square.6

But unlike ordinary number multiplication, forwhich a × b = b × a, the product ◦ does notalways commute. Compare:with7

DEFINITION. The symmetry operation thatfixes all points is called the identity operation.We denote the identity by the letter e.Examples:m1 ◦ m1 = em2 ◦ m2 = er ◦ r ◦ r ◦ r = r 4 = eDEFINITION. The number of times a symmetryoperation must be repeated to ”return” tothe identity is its order.The orders of m1 and m2 are 2; the order ofr is 4.8

Fill in the multiplication table for the symmetryoperations of the square.do this operation firstm2m1err2r3m1m1'm2 m2'eerr2eem1'rr3m1eem1'em2em2'em2'9

Your table should be identical with this one:m2m1'm2'm1rm2e err2r3r3do this operation firste r r2 r3 m1 m1' m2 m2'rr2rr2r3er2r3rm2m1m2' m1'm2' m2' m1' m2 m1r3 m1m1'm2 m2'rm1' m1m2'm2r2 m2' m2m1 m1m2'm1' m2 err3r2m1' m1' m2 m1m2'r2 eee m2 m2' m1' m1r3rm1 m1'r3rer2rr3r2e10

DEFINITION. The symmmetry operations ofan object constitute its symmetry group.DEFINITION. A group is a set G = {e, g 1 , g 2 , g 3 . . .}together with a product ◦, such thati) G is ”closed under ◦”: if g 1 and g 2 are anytwo members of G then so are g 1 ◦g 2 and g 2 ◦g 1 ;ii) G contains an identity e: for any g in G,e ◦ g = g ◦ e = g;iii) ◦ is associative: (g 1 ◦g 2 ) ◦g 3 = g 1 ◦(g 2 ◦g 3 );iv) Each g in G has an inverse g −1 that is alsoin G: g ◦ g −1 = g −1 ◦ g = e.Symmetry groups are a fundamental conceptof mathematical crystallography.11

If every element of G can can be written as aproduct of elements of some subset {g 1 , . . . , g k },then this subset generates G.Find a set of two generators for the symmetrygroup of the square.m2m1'm2'm1rm2e err2r3r3do this operation firste r r2 r3 m1 m1' m2 m2'rr2rr2r3er2r3rm2m1m2' m1'm2' m2' m1' m2 m1r3 m1m1'm2 m2'rm1' m1m2'm2r2 m2' m2m1 m1m2'm1' m2 err3r2m1' m1' m2 m1m2'r2 eee m2 m2' m1' m1r3rm1 m1'r3rer2rr3r2eFind another set of two generators for thisgroup.12

DEFINITION. The path of a point on an objectunder the action of its symmetry group Gis an orbit of G. Here the orbits of some pointsof the square.m2m1m1'rm2'13

Since a point lying on a symmetry element iskept in place by the corresponding symmetryoperation, the number of points in the orbit iscorrespondingly reduced.DEFINITION. The set S x of operations thatkeep a point S fixed is its site symmetry group.This terminology is appropriate because theoperations that fix a point satisfy the groupaxioms.S = {e}|S| = 1S = m2 or m2'|S| = 2S = G|S| = 8A little algebra shows that the number of pointsin the orbit of x is k, then k × |S| = |G| where|S| and |G| denote the corresponding numberof symmetry operations, including e.14