Lecture 4 Group Theory PDF

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Group Theory Abstract Algebra Mr. W Mangcengeza Lecture 4  Task Now that you know the properties by heart. Investigate the following: What is your unde...

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Group Theory Abstract Algebra Mr. W Mangcengeza Lecture 4  Task Now that you know the properties by heart. Investigate the following: What is your understanding of Group Theory in Abstract Algebra? How do we apply closure, associativity, identity and inverse property in Group Theory?  Definition of a Group A Group G is a collection of elements together with a binary operation* which satisfies the following properties: Closure Associativity Identity Inverses A binary operation is a function on G which assigns an element of G to each ordered pair of elements in G. For example, multiplication and addition are binary operations.  Classification of Groups Groups may be Finite or Infinite; that is, they may contain a finite number of elements, or an infinite number of elements Also, groups may be Commutative or Non-Commutative, that is, the commutative property may or may not apply to all elements of the group. Commutative groups are also called Abelian groups.  Examples of Groups Examples of Groups: Infinite, Abelian: The Integers under Addition (Z. +) The Rational Numbers without 0 under multiplication (Q*, X) Infinite, Non-Abelian: The non-singular n x n matrices under matrix multiplication Finite, Abelian: The Integers Mod n under Modular Addition (Zn , +)  Properties of a Group: Closure If we combine any two elements in the group under the binary operation, the result is always another element in the group. Example: The Integers under Addition, (Z, +) 1 and 2 are elements of Z, 1+2 = 3, also an element of Z Non-Examples: The Odd Integers are not closed under Addition. For example, 3 and 5 are odd integers, but 3+5 = 8 and 8 is not an odd integer. The Integers lack inverses under Multiplication, as do the Rational numbers (because of 0.) However, if we remove 0 from the Rational numbers, we obtain an infinite closed group under multiplication.  Properties of a Group: Associativity The Associative Property, states that. This may be extended to as many elements as necessary. For example: In Integers, Caution: In Matrix Multiplication, The Commutative Property, also familiar from ordinary (A*B)*C=A*(B*C). arithmetic on real numbers, does not generally apply to all In function composition, groups! Only Abelian groups are f*(g*h) = (f*g)*h. commutative. This is a property of all groups.  Properties of a Group: Identity The Identity Property, familiar from ordinary arithmetic on real numbers, states that, for all elements a in G, The Identity is Unique! For example, There is only one identity element in any group. in Integers, This property is used in proofs. In (Q*, X), a*1 = 1*a = a. In Matrix Multiplication, A*I = I*A = A. This is a property of all groups.  Properties of a Group: Inverses The inverse of an element, combined with that element, gives the identity. Inverses are unique. That is, each element has exactly one inverse, and no two distinct elements have the same inverse. The uniqueness of inverses is used in proofs. For example... In (Z,+), the inverse of is. In (Q*, X), the inverse of is. In abstract algebra, the inverse of an element a is usually written  Abelian Groups Abelian Groups are groups which have the Commutative property, a*b = b*a for all a and b in G Abelian groups are named after Neils Abel, a Norwegian mathematician. Cayley tables for Z4 and U8 Abelian groups may be recognized by a diagonal symmetry in their Cayley table (a table showing the group elements and the results of their composition under the group binary operation.) This symmetry may be used in constructing a Cayley table, if we know that the group is Abelian.  Activity The binary operator multiplication modulo 14, denoted by *, is defined on the set: Copy and * complete 2 4 6 the 8following 10 12 4 8 12 6 operation 2 table. 2 10 4 8 2 10 4 12 6 6 12 10 8 6 4 2 8 2 4 6 8 10 12 10 6 12 4 10 2 8 12 10 6 2 12 8 4  Activity Show that is a group (it satisfies the requirements of closure, associativity, identity and inverse) * 2 4 6 8 10 12 2 4 8 12 2 6 10 4 8 2 10 4 12 6 6 12 10 8 6 4 2 8 2 4 6 8 10 12 10 6 12 4 10 2 8 12 10 6 2 12 8 4  Closure G is closed under * for all S, * is closed since all members of the Cayley table are in S.  Associativity * is associative in G for all S, * is associative since multiplication modulo n is.  Identity 8 is the identity element G must contain the identity element of operation * There exists such that for all  Inverses Here means that the inverse of 2 is 4 For each there exists such that (because when you combine them you get the identity which is 8 in this case).. Now find the inverse of 4, 6, For each , exists as shown. 8, 10