242 DETERMINANTS.MATRICES.ANDGROUP THEORY DlSCRETEGROUPS 243

Consider a subgroup H with elements hi and a group eiement x not in H.Then Xhi and hix are not in subgroup H. The sets generated by

xhi i = 1,2, ... and hix i = 1,2, ...{

X-X4. rctlcction

y--y.

(a) Show that these four operations form a group.(b) Show that this group is isomorphic with the vierergruppe.(c) Set up a 2 x 2 matrix representation.

4.8.4 Rearrangement theorem.Given a group of n distinct elements (1,a, b, c, . . . ,n), showthat the setof products(aI. a2, ab, ac.. . . .ali) reproduces the 11distinct elements in a new order.

4.8.5 Using the 2 x 2 matrix representation of Exercise 4.2.7 for the vierergruppe,(a) Show that there are four classes, each with one element.(b) Calcula te the character (trace) of each class. Note that two different classes

may have the same character.(c) Show that there are three two-element subgroups. (fhe uni! element by

itself always forms a subgroup.)(d) For any one of the two-element subgroups show that the subgroup and a

singIe coset reproduce the original vierergruppe.Note that subgroups, classes. and cosets are entirely different.

4.8.6 Using the 2 x 2 matrix representation, Eq. 4.208, of C4,(a) Show that there are four classes, each with one element.(b) Calculate the character (trace) of each class.(c) Show that there is one two-element subgroup.(d) Show that the subgroup and a single coset reproduce the original group.

4.8.7 Prove that the number of distinct elements in a coset of a subgroup is the sameas the number of elements in the subgroup.

4.8.8 A subgroup H has elements hi. x is a fixed element of the original group G andis no/ a member of H. The transform

xhix-I i= 1,2, ...

generates a conjugate subgroupxH X-I. Show that this conjugate subgroup satisfieseach ofthe four group postulates and therefore is a group.

are called cosels, res~ctively, the left and right cosets of subgroup H withrespect to x. It can be slÍown (assume the contrary and prove a contradiction)that the coset of a subgroup has the same number of distinct elements as thesubgroup. Extending this result we may express the original group G as the sumor H and cosets:

G=H+x!H+X2H+ ....

Then lhe order of an)' subgroup is a divisor of lhe order of lhe group. It is thisresult that makes the concept of coset significant. In the next section the six-element group D3 (order 6) has subgroups of order 1,2, and 3. D3 cannot (anddoes not) have subgroups of order 4 or 5.

The similarity transform of a subgroup H by a fixed group element x nol inH, xHx-! yields a subgroup-Exercise 4.8.8. If this new subgroup is identicalwith H for ali x,

xHx-! = H,

then H is called an invarianl, normal, or self-conjugale subgroup. Such subgroupsare involved in the analysis of multiplets of atomic and nuclear spectra and theparticles discussed in Section 4.12. Ali subgroups of a commutative (abelian)group are automatically invariant.

EXERCISES

4.8.1 Show that the matrices 1, A, B, and C of Eq. 4.208 are reducible. Reduce them.Note. This means transforming A and C to diagonal form (by the same unitarytransformation).Hint. A and C are anti-Hermitian. Their eigenvectors will be orthogonal.

4.8.2 Possible operations on a crystallattice include A. (rotation by 1t),m (retlection),and i (inversion). These three operations combine as

A~ = m2 = i2 = I,

4.8.9 (a) A particular group is abelian. A second group is created by replacing gi bygil for each element in the original group. Show that the two groups areisomorphic.No/e. This means showing that if aibi = Ciothen, ailbil = cil.Continuing part (a), if the two groups are isomorphic, show that each mustbe abelian.

(b)

A1[.n1= i, m.i=A1f' and i.A.=m.

Show that the group (I,A.,m, i) is isomorphic with the vierergruppe.4.9 DISCRETE GROUPS

4.8.3 Four possible operations in the xy-plane are:

I. no change{

X - x

)'-.1'

{

X - -x2. inversion

.1'- -)'

3. retlection {X - - x

y-y

In physics, groups usually appear as a set or operations that leave a systemunchanged, invariant. This is an expression of symmetry. Indeed, a symmetrymay be defined as the invariance ofthe Hamiltonian of a system ul)der a grouportransformations. Symmetry in this sense is important in cIassical mechanics,but it becomes even more important and more profound in quantum mechanics.In this section we investigate the symmetry properties or sets of objects (atomsin 'a molecule or crystal). This provides additional illustrations or the group

250 DETERMINANTS.MATRICES.ANDGROUP THEORY CONTINUOUS GROUPS 251

(O I O

)(a

) (b

)001 b=c.

I O O c a

4.9.10 The permutation group of four objects contains 4! = 24 elements. From Ex.4.9.9, D4' the symmetry group for a square, has far less than 24 elements. Explainthe relation between D4 and the permutation group of four objects.

4.9.11 A plane is covered with regular hexagons, as shown.(a) Determine the dihedral symmetry of an axis perpendicular to the plane

through the common vertex of three hexagons (A). That is, if the axis hasII-fold symmetry, show (with careful explanation) what 11is. Write out the2 x 2 matrix describing the minimum (nonzero) positive rotation of the

array of hexagons that is a member of your D. group.(b) Repcat part (a) for an axis perpendicular to the plane through the geometric

center of one hexagon (B).

so on. As a permutation (a b c) -> (b c a). In three dimensions

(a) Develop analo,gous' 3 x 3 representations for the other elements of D3'(b) Reduce your,3 x '3 representation to the 2 x 2 representation ofthis section.(This 3 x 3 reprcsentation mus/ be reducible or Eq. 4.220 would be violated.)No/e. The actual reduction of a reducible representation may be awkward. Itis often easier to develop directly a new representation of the required dimension.

4.9.5 (a) The permutation group of four objects, P4' has 4! = 24 elements. Treatingthe four elements of the cyclic group, C4, as permutations, set up a 4 x 4matrix representation of C4. C4 becomes a subgroup of P4.

(b) How do you know that this 4 x 4 matrix representation of C4 mus/ bereducible?

No/e. C4 is abelian and every abelian group of h objects has only h one-dimensional irreducible representations.

4.9.6 (a) The objects (a b c d) are permuted to (d a c h). Write out a 4 x 4 matrixrepresentation of this one permutation.

(b) Is permutation, (a h d c) -> (d a c b), odd or evcn?(c) Is this permutation a possible member of the D4 group? Why or why not?

4.9.7 The elements of the dihedral group D. may be written in the form

S)'R~(21t/II), }, = O, I

p=O, I, ...,11-1,

where R:(21t/II) represents a rotation of 21t/1I about the II-fold symmetry axis,whereas S represents a rotation of 1t about an axis through the center of theregular polygon and one of its vertices.For S = E show that this form may describe the matrices A, B, C, and O of DJ'

No/e. The elements R: and Sare called the gellera/ors of this finite group.Similarly, i is the generator of the group given by Eq. 4.207.

4.9.8 Show that the cyclic group of 11objects, C., may be represented by rm,m = O, I,2, . . . ,11- I. Here r is a generator given hy

,. = exp(21tiS/II).

The parameter s takes on the values s = 1,2,3, ...,11, each value of s yieldinga differept one-dimensional (irreducible) representation of C..

4.9.9 Develop the irreducible 2 x 2 matrix representation of the group of operations(rotations and reOections) that transform a square into itself. Give the groupmultiplication table.No/e. This is the symmetry group of a square and also the dihedral group, D4'

4.9.12 In a simple cubic crystal, we might have identical atoms at r = (Ia,ma,lia), I,m, and 11taking on all integral values.(a) Show that each cartcsian axis is a fourfold symmetry axis.(b) The cllbic group will consist of ali operations (rotations, reOections, in-

version) that leave the simple cubic crystal invariant. From a considerationof the permutation of the positive and negative coordinate axes, predicthow many elements this cubic group will contain.

From the DJ multiplication table construct a similarity transform tableshowing xyx-I, where x and y each range over all six elements of DJ:

4.9.13 (a)

~x

y 1 A---

1 1 1 ---

A A A---

(b) Divide the elements of DJ into classes. Using the 2 x 2 matrix representationof Eqs. 4.215 to 4.218 note the trace (character) of each class.

4.10 CONTINUOUS GROUPSl'

Infinite Groups. Lie Groups

. Ali of the groups in the two preceding sections have contained a finite num-ber óf elements: four for the vierergruppe, six for D3, and so on. Here we intro-