Mathematical Methods for Physics and Engineering - Matematica.NET

28.5 MAPPINGS BETWEEN GROUPS28.5 Mappings between groupsNow that we have available a range of groups that can be used as examples,we return to the study of more general group properties. From here on, whenthere is no ambiguity we will write the product of two elements, X • Y , simplyas XY , omitting the explicit combination symbol. We will also continue to use‘multiplication’ as a loose generic name for the combination process betweenelements of a group.If G and G ′ are two groups, we can study the effect of a mappingΦ:G → G ′of G onto G ′ .IfX is an element of G we denote its image in G ′ under the mappingΦbyX ′ =Φ(X).A technical term that we have already used is isomorphic. We will now defineit formally. Two groups G = {X,Y,...} and G ′ = {X ′ ,Y ′ ,...} are said to beisomorphic if there is a one-to-one correspondencebetween their elements such thatX ↔ X ′ ,Y↔ Y ′ , ···XY = Z implies X ′ Y ′ = Z ′and vice versa.In other words, isomorphic groups have the same (multiplication) structure,although they may differ in the nature of their elements, combination law andnotation. Clearly if groups G and G ′ are isomorphic, and G and G ′′ are isomorphic,then it follows that G ′ and G ′′ are isomorphic. We have already seen an example offour groups (of functions of x, of orthogonal matrices, of permutations and of thesymmetries of an equilateral triangle) that are isomorphic, all having table 28.8as their multiplication table.Although our main interest is in isomorphic relationships between groups, thewider question of mappings of one set of elements onto another is of someimportance, and we start with the more general notion of a homomorphism.Let G and G ′ be two groups and Φ a mapping of G → G ′ . If for every pair ofelements X and Y in G(XY ) ′ = X ′ Y ′then Φ is called a homomorphism, and G ′ is said to be a homomorphic image of G.The essential defining relationship, expressed by (XY ) ′ = X ′ Y ′ , is that thesame result is obtained whether the product of two elements is formed first andthe image then taken or the images are taken first and the product then formed.1059