Mathematical Methods for Physics and Engineering - Matematica.NET

28.7 SUBDIVIDING A GROUPthis implies that Z belongs to S Y . These two results together mean that the twosubsets S X and S Y have the same members and hence are equal.Now suppose that S X equals S Y .SinceY belongs to S Y it also belongs to S Xand hence X ∼ Y . This completes the proof of (i), once the distinct subsets oftype S X are identified as the classes C i . Statement (ii) is an immediate corollary,the class in question being identified as S W .The most important property of an equivalence relation is as follows.Two different subsets S X and S Y can have no element in common, and the collectionof all the classes C i is a ‘partition’ of S, i.e. every element in S belongs to one, andonly one, of the classes.To prove this, suppose S X and S Y have an element Z in common; then X ∼ Zand Y ∼ Z and so by the symmetry and transitivity laws X ∼ Y .Bytheabovetheorem this implies S X equals S Y . But this contradicts the fact that S X and S Yare different subsets. Hence S X and S Y can have no element in common.Finally, if the elements of S are used in turn to define subsets and hence classesin S, every element U is in the subset S U that is either a class already found orconstitutes a new one. It follows that the classes exhaust S, i.e.everyelementisin some class.Having established the general properties of equivalence relations, we now turnto two specific examples of such relationships, in which the general set S has themore specialised properties of a group G and the equivalence relation ∼ is chosenin such a way that the relatively transparent general results for equivalencerelations can be used to derive powerful, but less obvious, results about theproperties of groups.28.7.2 Congruence and cosetsAs the first application of equivalence relations we now prove Lagrange’s theoremwhich is stated as follows.Lagrange’s theorem. If G is a finite group of order g and H is a subgroup of G oforder h then g is a multiple of h.We take as the definition of ∼ that, given X and Y belonging to G, X ∼ Y ifX −1 Y belongs to H. This is the same as saying that Y = XH i for some elementH i belonging to H; technically X and Y are said to be left-congruent with respectto H.This defines an equivalence relation, since it has the following properties.(i) Reflexivity: X ∼ X, sinceX −1 X = I and I belongs to any subgroup.(ii) Symmetry: X ∼ Y implies that X −1 Y belongs to H and so, therefore, doesits inverse, since H is a group. But (X −1 Y ) −1 = Y −1 X and, as this belongsto H, it follows that Y ∼ X.1065