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GROUP THEORYThree immediate consequences of the above definition are proved as follows.(i) If I is the identity of G then IX = X for all X in G. ConsequentlyX = (IX) = I X ,for all X in G . Thus I is the identity in G . In words, the identity elementof G maps into the identity element of G .(ii) Further,I = (XX −1 ) = X (X −1 ) .That is, (X −1 ) = (X ) −1 . In words, the image of an inverse is the sameelement in G as the inverse of the image.(iii) If element X in G is of order m, i.e. I = X m , thenI = (X m ) = (XX m−1 ) = X (X m−1 ) = · · · = X X · · · X .m factorsIn words, the image of an element has the same order as the element.What distinguishes an isomorphism from the more general homomorphism arethe requirements that in an isomorphism:(I) different elements in G must map into different elements in G (whereas ina homomorphism several elements in G may have the same image in G ),that is, x = y must imply x = y;(II) any element in G must be the image of some element in G.An immediate consequence of (I) and result (iii) for homomorphisms is thatisomorphic groups each have the same number of elements of any given order.For a general homomorphism, the set of elements of G whose image in G is I is called the kernel of the homomorphism; this is discussed further in thenext section. In an isomorphism the kernel consists of the identity I alone. Toillustrate both this point and the general notion of a homomorphism, considera mapping between the additive group of real numbers and the multiplicativegroup of complex numbers with unit modulus, U(1). Suppose that the mapping → U(1) isthen this is a homomorphism sinceΦ : x → e ix ;(x + y) → e i(x+y) = e ix e iy = x y .However, it is not an isomorphism because many (an infinite number) of theelements of have the same image in U(1). For example, π, 3π, 5π, . . . in allhave the image −1 in U(1) and, furthermore, all elements of of the form 2πn,where n is an integer, map onto the identity element in U(1). The latter set formsthe kernel of the homomorphism.1060
28.6 SUBGROUPS(a)I A B C D EI I A B C D EA A B I E C DB B I A D E CC C D E I A BD D E C B I AE E C D A B I(b)I A B CI I A B CA A I C BB B C I AC C B A ITable 28.9 Reproduction of (a) table 28.8 and (b) table 28.3 with the relevantsubgroups shown in bold.For the sake of completeness, we add that a homomorphism for which (I) aboveholds is said to be a monomorphism (or an isomorphism into), whilst a homomorphismfor which (II) holds is called an epimorphism (or an isomorphism onto). If,in either case, the other requirement is met as well then the monomorphism orepimorphism is also an isomorphism.Finally, if the initial and final groups are the same, G = G , then the isomorphismG → G is termed an automorphism.28.6 SubgroupsMore detailed inspection of tables 28.8 and 28.3 shows that not only do thecomplete tables have the properties associated with a group multiplication table(see section 28.2) but so do the upper left corners of each table taken on theirown. The relevant parts are shown in bold in the tables 28.9(a) and (b).This observation immediately prompts the notion of a subgroup. A subgroupof a group G can be formally defined as any non-empty subset H = {H i } ofG, the elements of which themselves behave as a group under the same rule ofcombination as applies in G itself. As for all groups, the order of the subgroup isequal to the number of elements it contains; we will denote it by h or |H|.Any group G contains two trivial subgroups:(i) G itself;(ii) the set I consisting of the identity element alone.All other subgroups of G are termed proper subgroups. In a group with multiplicationtable 28.8 the elements {I, A, B} form a proper subgroup, as do {I, A} in agroup with table 28.3 as its group table.Some groups have no proper subgroups. For example, the so-called cyclicgroups, mentioned at the end of subsection 28.1.1, have no subgroups otherthan the whole group or the identity alone. Tables 28.10(a) and (b) show themultiplication tables for two of these groups. Table 28.6 is also the group tablefor a cyclic group, that of order 4.1061
Page 2: GROUP THEORYNHMHHH(a)(b)HFigure 28.
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29.11 PHYSICAL APPLICATIONS OF GROU
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29.12 EXERCISESas the sum of two on
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29.12 EXERCISESUse this to show tha
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29.13 HINTS AND ANSWERS(a) Make an
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Group theory

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