Mathematical Methods for Physics and Engineering - Matematica.NET

28.1 GROUPS28.1.1 Definition of a groupAgroupG is a set of elements {X,Y,...}, together with a rule for combiningthem that associates with each ordered pair X, Y a ‘product’ or combination lawX • Y for which the following conditions must be satisfied.(i) For every pair of elements X, Y that belongs to G, the product X • Y alsobelongs to G. (This is known as the closure property of the group.)(ii) For all triples X, Y , Z the associative law holds; in symbols,X • (Y • Z) =(X • Y ) • Z. (28.1)(iii) There exists a unique element I, belonging to G, with the property thatI • X = X = X • I (28.2)for all X belonging to G. This element I is known as the identity elementof the group.(iv) For every element X of G, there exists an element X −1 , also belonging toG, such thatX −1 is called the inverse of X.X −1 • X = I = X • X −1 . (28.3)An alternative notation in common use is to write the elements of a group Gas the set {G 1 ,G 2 ,...} or, more briefly, as {G i }, a typical element being denotedby G i .It should be noticed that, as given, the nature of the operation • is not stated. Itshould also be noticed that the more general term element, rather than operation,has been used in this definition. We will see that the general definition of agroup allows as elements not only sets of operations on an object but also sets ofnumbers, of functions and of other objects, provided that the interpretation of •is appropriately defined.In one of the simplest examples of a group, namely the group of all integersunder addition, the operation • is taken to be ordinary addition. In this group therole of the identity I is played by the integer 0, and the inverse of an integer X is−X. That requirements (i) and (ii) are satisfied by the integers under addition istrivially obvious. A second simple group, under ordinary multiplication, is formedby the two numbers 1 and −1; in this group, closure is obvious, 1 is the identityelement, and each element is its own inverse.It will be apparent from these two examples that the number of elements in agroup can be either finite or infinite. In the former case the group is called a finitegroup and the number of elements it contains is called the order of the group,which we will denote by g; an alternative notation is |G| but has obvious dangers1043