Group, Abelian element

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

Prove that in any Abelian group, each element is in a class by itself. 

As in point groups, the number of classes equals the number of irreducible representations F of the group A. In Abelian groups any element forms a dass on its own. Consequently, In Abelian groups the number of dasses is identical to thdr order. Abelian groups thus... 

Since AiG g) 1 Abelian group, any element is a class on its own. Consequently, AiG ) has four irredudble representations, denoted as Fo to Fa. The characters of the elements of AiG g) he individual irredudble representations are given in Table 4, which is the character table of the group A(Gig)> For determining the characters X(Aj,Fjk) reference should again be made to the literature on group theory. 

We will denote by Z the group whose elements are the lattice points of Euclidean M-space R", with the group operation being vector addition it is a free abelian group of rank , with (1,0... 0), (0,1,0..., 0), (0,..., 0,1) as a basis. 

For an Abelian group every element commutes with every other, thus the group must be represented by diagonal matrix representations since only sets of diagonal matrices will, in general, commute. The representations must therefore be one-dimensional because a set of diagonal matrices of dimension greater than one.will commute with a nonconstant matrix, and must therefore be reducible. 

The identity element is clearly in a class by itself. It is also the only class that is a subgroup. All other classes lack the identity element. In an Abelian group, all elements commute, and every element is therefore in a class by itself... 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

The product of group elements is not necessarily commutative and in general AB BA. If all the elements of a group commute with each other, it is said to be an Abelian group. 

Consider the set of rotations of a circle about an axis normal to the plane of the circle and passing through its centre. Each element of this set is characterized by one parameter which may be chosen to be the angle of rotation (/> which varies in the interval [0, 27r]. This is a one-parameter, continuous, connected, abelian, compact Lie group, known as the axial rotation group, denoted by 0(2). 

It is dear that the elements of point groups do not necessarily commute, that is the order in which one combines two symmetry operations can be important (see, for example, Fig. 2-4.1 and Table 3-4.1 where for the symmetric tripod Ctar j o C ). A group for which all the elements do commute is called an Abelian group. 

Note the convention that in forming each product, the element on the side of the table is put on the left. In the definition of a group, it was not postulated that AB = BA. Groups such that AB = BA for all pairs of elements are called commutative or Abelian. 

An Abelian group has each element in a class by itself. The converse of this theorem is also true A group with each element in a class by itself is Abelian. The proof is simple By hypothesis C 1AC=A for all elements A and C left multiplication by C gives AC—CA. Q.E.D. The number of symmetry operations in each class of a point group is indicated by an integer in front of the symbol for the symmetry operation, and it is therefore easy to see whether a group is Abelian by looking at the top line of the character table. 

There are some groups, however, in which combination is commutative, and such groups are called Abelian groups. Because of the fact that multiplication is not in general commutative, it is sometimes convenient to have a means of stating whether an element B is to be multiplied by A in the sense AB or BA. In the first case we can say that B is left-multiplied by A, and in the second case that B is right-multiplied by A. 

Let us consider now the cases where the only symmetry element is a proper axis, C . This generates a set of operations C , CJ, Cj,.. . , Cna = E. Hence a molecule with C as its only symmetry element would belong to a group of order , which is designated C . It may be noted that a C group is a cyclic group (see Section 2.2) and hence also Abelian. 

As noted earlier, a cyclic group is Abelian, and each of its h elements is in a separate class. Therefore, it must have h one-dimensional irreducible representations. To obtain these there is a perfectly general scheme which is perhaps best explained by an example. It will be evident that the example may be generalized. Let us consider the group C5, consisting of the five commuting operations C5, C, C, C5, E we seek a set of five one-... 

G and H are said to be conjugate to one another. A subset of G whose members are all conjugate to one another is termed a class. Note that the identity is in a class by itself, so that no class can be a proper subgroup of G It is also obvious from Eq. 1.10 that if G is Abelian each element is in a different class. The group of order six whose multiplication table was given previously has three classes formed by the elements E A,B and C,D,F as can be seen by inspection. 

Exercise 1.1-2 (a) Show that cyclic groups are Abelian, (b) Show that for a finite cyclic group the existence of the inverse of each element is guaranteed, (c) Show that uj cxp( 2n /n) generates a cyclic group of order n, when binary composition is defined to be the multiplication of complex numbers. 

Exercise 1.8-2 (a) Z(C(3)) is the set of elements of C(3) that commute with every element of C(3). From Table 1.3 we see that each element of C(3) commutes with every other element (the mutiplication table of C(3) is symmetrical about its principal diagonal from upper left to lower right) so that Z(C(3)) = C(3), and consequently C(3) is an Abelian group. 

Prove that if each element of a group G commutes with every other element of G (so that G is an Abelian group) then each element of G is in a class by itself. 

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