The factor group

Suppose that H is an invariant subgroup of G of index t. Then the t cosets gr H of H (including y, H = H) each considered as one element, form a group of order t called the 

Each term in parentheses, gr FL, is one element of F. Because each element of F is a set of elements of G, binary composition of these sets needs to be defined. Binary composition of the elements of F is defined by 

Exercise 1.4-1 Show that g g = g is both a necessary and sufficient condition for gi to be E, the identity element in G. [Hint Recall that the identity element E is defined by 

F contains the identity that F is indeed a group requires the demon-stration of the validity of the other group properties. These follow from the definition of binary composition in F, eq. (2), and the invariance of H in G. 

Closure To demonstrate closure we need to show that gpgq H e F for gp, gq, g, C gr.  

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Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

Table 2 Correlation of the molecular point group of Sg with the factor group of the orthorhombic crystal (E>4d—>C2— D2h) [88]...

The factor group Dzh of orthorhombic Sg includes an inversion operation therefore, the g-u exclusion principle works resulting in modes of either Raman (gerade, g) or infrared activity (ungerade, u). 

The V3 mode observed at about 415 cm is inactive in the free molecule and, therefore, the factor group components are very weak in the Raman... 

This is a sub-group of the space group, from which the effects of simple translations have been factored off . Isomorphous means here isomorphous to the factor group . 

Methods for treating the factor group vibrations have been given by Davydov (25), as well as by Bhagavantam and Venkatarayudu (22). A simple analysis is possible through what is known as the correlation method (20, 26, 27) by which one is able to write the irreducible representations and thereby classify k=0 phonons directly and simply. The number of A = 0 phonons is 3 N, where N is taken to be the number of atoms in the entire unit cell. However, there are only 3 N-3 optically active phonons because the acoustic vibrations have approximately... 

The 17 plane groups are not mutually unrelated. Some of them are subgroups of other plane groups, as shown in Fig. E.l. The order of the factor group, that is, the number of different symmetry operations other than translational symmetry, is also shown for each plane group. 

Fig. E.l. Relations among plane groups. In this figure, the plane groups are shown in their degrees of symmetry, as indicated by the order of the factor groups. A plane group with high symmetry always has one or several subgroup(s). The chart shows such relations within the same lattice.

Second, a multiplication table for the factor group is written down. The space group formed by the above symmetry elements is infinite, because of the translations. If we define the translations, which carry a point in one unit cell into the corresponding point in another unit cell, as equivalent to the identity operation, then the remaining symmetry elements form a group known as the factor, or unit cell, group. The factor... 

In these equations, m is the mass of an element (a CHa group), kc is the force constant for stretching of a C—C bond, ka is the force constant for bending of the CCC angle, and

phase difference between the motions of adjacent elements of the chain. Since we are interested only in the factor group modes, i.e those in which the vibrations of corresponding elements in neighboring unit cells are in phase, we require that... 

Exercise 1.4-3 Show that, with binary composition as multiplication, the set 1 —1 i —i, where i2 = — 1, form a group G. Find the factor group F = G/H and write down its multiplication table. Is F isomorphous with a permutation group ... 

Equation (10) confirms that F(k) is a group and that F(k) P(k). In mappings of the factor group view of the... 

Table 18.2. Character table of the factor group G(q)/T at A [0 q 0], together with the corresponding classes of G(q)/T at T and the Jones symbols R(xyz), where (xyz) is an abbreviation for (ex ey ez).

Table 18.4. Corresponding classes of the factor group G/T at T and A in the BZ of silicon and characters for the direct sums at A that are compatible with IRs at T.

Both methods can easily be derived from a very simple model. Consider a unit cell which contains two XY molecules which are equivalent in the group-theoretical sense, i.e. they are transformed into one-another by the operations of the group of the unit cell (this group is the factor group of the space group 10 and is isomorphous with one of the 32 crystallographic point groups).8 ... 

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