《Discrete Mathematics II教学-华南理工》Abstract Algebra I.pptVIP

Abstract Algebra I?Academic year 2010-2011 Contents: Topics in Group Theory 1.1 Groups 1.2 Examples of Groups 1.3 Elementary Properties of Groups 1.1 Groups A binary operation ? on a set G associates to elements x and y of G a third element x ? y of G. For example, addition and multiplication are binary operations of the set of all integers. Definition: A group G consists of a set G together with a binary operation ? for which the following properties are satisfied: ? (x ?y) ? z = x ?(y ?z) for all elements x, y, and z of G (the Associative Law ); ? There exists an element e of G (known as the identity element of G) such that e ?x = x = x ?e, for all elements x of G; ? For each element x of G there exists an element x -1 of G (known as the inverse of x) such that x ?x = e = x ?x (where e is the identity element of G). The order |G| of a finite group G is the number of elements of G. A group G is Abelian (or commutative) if x ?y = y ?x for all elements x and y of G. One usually adopts multiplicative notation for groups, where the pro duct x ?y of two elements x and y of a group G is denoted by xy. The associative property then requires that (xy)z = x(yz) for all elements x, y and z of G. The identity element is often denoted by e (or by e when it is necessary G to specify explicitly the group to which it belongs), and the inverse of an element x of G is then denoted by x -1 It is sometimes convenient or customary to use additive notation for certain groups. Here the group operation is denoted by +, the identity element of the group is denoted by 0, the inverse of an element x of the group is denoted by x. By convention, additive notation is rarely used for non-Abelian groups. When expressed in additive notation the axioms for a Abelian group require that (x + y ) + z = x + (y + z), x + y = y + x, x + 0 = 0 + x = x and x + (? x) = (? x) + x = 0 for all elements x, y and z of the group. We shall usually employ multiplicative notatio