Mathematics Course 111: Algebra I Part II: Groups
نویسنده
چکیده
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′ 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 product x ∗ y of two elements x and y of a group G is denoted by xy. The inverse of an element x of G is then denoted by x−1. The identity element is usually denoted by e (or by eG when it is necessary to specify explicitly the group to which it belongs). Sometimes the identity element is denoted by 1. Thus, when multiplicative notation is adopted, the group axioms are written as follows:• (xy)z = x(yz) 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 ex = x = xe, 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 xx−1 = e = x−1x (where e is the identity element of G). The group G is said to be Abelian (or commutative) if xy = yx for all elements x and y of G. 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 only used for Abelian groups. When expressed in additive notation the axioms for a Abelian group are as follows: • x+ y = y + x for all elements x and y of G (the Commutative Law);
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