Group, Abelian classes

H2i X) is the free abelian group generated by the homology classes of the closures of the i-dimensional cells. 

The present review is based mainly on our publications [33,35-39,49-53]. In Section II we give a detailed description of the general reduction routine for an arbitrary relativistically invariant systems of partial differential equations. The results of Section II are used in Section III to solve the problem of symmetry reduction of Yang-Mills equations (1) by subgroups of the Poincare group P 1,3) and to construct their exact (non-Abelian) solutions. In Section IV we review the techniques for nonclassical reductions of the STJ 2) Yang-Mills equations, which are based on their conditional symmetry. These techniques enable us to obtain the principally new classes of exact solutions of (1), which are not derivable within the framework of the standard symmetry reduction technique. In Section V we give an overview of the known invariant solutions of the Maxwell equations and construct multiparameter families of new ones. 

Consider an Abelian group of order h. Since the group is Abelian, it has h classes (Section 9.2) and therefore h irreducible representations. Theorem... 

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. 

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. 

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

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. 

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. 

The group C3 is Abelian and has three classes there are therefore three IRs and each IR occurs once in Tr. (But note that the matrices of Tr are not block-diagonal.)... 

Let C be an abelian category. Following lONEDA one can define the functors Eg(-,-) =E (-,-) of n-fold extension groups (without the use of Injective or projective objects In C), For A,Be C 2ui element of E (A,B) Is the equivalence class containing exact sequence... 

Actually, all the groups used in the following are Abelian. Thus for each subgroup // of G, a factor group G/H is defined. The function that maps each element of G to its class in G/H is called the canonical mapping. 

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. 

This result tells us that, in a group of order G, there are exactly G independent vectors. Rewriting these vectors in the form of SALCs exhausts all possible symmetries that can be realized in this group. If a group is abelian, every class is a singleton, and hence the number of classes is equal to the order of the group. In this case, Eq. (4.43) can be fulfilled only if all irreps are one-dimensional. Hence, in an abelian group all irreps are one-dimensional. 

The existence of at least one invariant mean characterizes a class of groups, the amenable groups, which include translation groups R", rotation groups O", and Euclidian groups E . Rem. the reason is R" abelian, G compact, and E semi-direct product of R and O") However, the Loren tz group is not amenable. 

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