Jeanine “Joni” Pinkney Central Washington University Abstract Algebra Part 1 Cumulative Review Text: Contemporary Abstract Algebra by J. A. Gallian, 6th edition This presentation by: Jeanine “Joni” Pinkney in partial fulfillment of requirements of Master of Arts in Mathematics Education degree Central Washington University Fall 2008 Picture credit: euler totient graph http://www.123exp-math.com/t/01704079357/

Contents: Chapter 2. Groups Definition and Examples Elementary Properties Chapter 3: Finite Groups; Subgroups Terminology and Notation Subgroup Tests Examples of Subgroups Chapter 4: Cyclic Groups Properties of Cyclic Groups Classifications of Subgroups of Cyclic Groups Chapter 5: Permutation Groups Definition and Notation Cycle Notation Properties of Permutations Suggested Activities Practice with Cyclic Notation Online Resources provided by text author J.A. Gallian Other Online Resources Acknowledgments Photo credit:A5, the smallest nonabelian group http://www.math.metu.edu.tr/~berkman/466object.html

Suggested Uses of this Presentation: Review for final exam for Math 461* Review in preparation for Math 462* Review for challenge exam for course credit for Math 461* Independent Study *or similar course math cartoons from http://www.math.kent.edu/~sather/ugcolloq.html

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. rubic cube permutation group http://en.wikipedia.org/wiki/Permutation_group

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. “Abelian... Isn't that a one followed by a bunch of zeros?” - anonymous grad student in MAT program symmetry 6 ceiling art http://architecture-buildingconstruction.blogspot.com/2006_03_01_archive.html

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 General Linear Groups (GL,n), the nonsingular nxn matrices under matrix multiplication Finite, Abelian: The Integers Mod n under Modular Addition (Zn , +) The “U groups”, U(n), defined as Integers less than n and relatively prime to n, under modular multiplication. Finite, Non-Abelian: The Dihedral Groups Dn the permutations on a regular n-sided figure under function composition. The Permutation Groups Sn, the one to one and onto functions from a set to itself under function composition. euler totient graph http://www.123exp-math.com/t/01704079357/

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.” -- Geoff “Not necessarily another element of the group!” -- Joni 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. "members only" http://en.wikipedia.org/wiki/index.html?curid=12686870

Properties of a Group: Associativity The Associative Property, familiar from ordinary arithmetic on real numbers, states that (ab)c = a(bc). This may be extended to as many elements as necessary. For example: In Integers, a+(b+c) = (a+b)+c. In Matrix Multiplication, (A*B)*C=A*(B*C). In function composition, f*(g*h) = (f*g)*h. This is a property of all groups. Caution: The Commutative Property, also familiar from ordinary arithmetic on real numbers, does not generally apply to all groups! Only Abelian groups are commutative. This may take some “getting-used-to,” at first! associative loop http://en.wikipedia.org/wiki/List_of_algebraic_structures

Properties of a Group: Identity The Identity Property, familiar from ordinary arithmetic on real numbers, states that, for all elements a in G, a+e = e+a = a. For example, in Integers, a+0 = 0+a = a. In (Q*, X), a*1 = 1*a = a. In Matrix Multiplication, A*I = I*A = A. This is a property of all groups. The Identity is Unique! There is only one identity element in any group. This property is used in proofs. |1 0| |0 1| = I

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 x is -x. In (Q*, X), the inverse of x is 1/x. In (Zn, +), the inverse of x is n-x. In abstract algebra, the inverse of an element a is usually written a-1. This is why (GL,n) and (SL, n) do not include singular matrices; only nonsingular matrices have inverses. In Zn, the modular integers, the group operation is understood to be addition, because if n is not prime, multiplicative inverses do not exist, or are not unique. The U(n) groups are finite groups under modular multiplication.

Cayley tables for Z4 and U8 Abelian Groups Abelian Groups are groups which have the Commutative property, a*b=b*a for all a and b in G. This is so familiar from ordinary arithmetic on Real numbers, that students who are new to Abstract Algebra must be careful not to assume that it applies to the group on hand. Abelian groups are named after Neils Abel, a Norwegian mathematician. Neils Abel postage stamp http://en.wikipedia.org/wiki/Neils_Abel 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. Cayley tables for Z4 and U8 http://www.math.sunysb.edu/~joa/MAT313/hw-VIII---313.html

Examples of Abelian Groups Some examples of Abelian groups are: The Integers under Addition, (Z,+) The Non-Zero Rational Numbers under Multiplication, (Q*, X) The Modular Integers under modular addition, (Zn, +) The U-groups, under modular multiplication, U(n) = {the set of integers less than or equal to n, and relatively prime to n} All groups of order 4 are Abelian. There are only two such groups: Z4 and U(4). http://www.math.csusb.edu/faculty/susan/modular/modular.html

Non-Abelian Groups Some examples of Non-Abelian groups are: Dn, the transformations on a regular n-sided figure under function composition (GL,n), the non-singular square matrices of order n under matrix multiplication (SL,n), the square matrices of order n with determinant = 1under matrix multiplication Sn, the permutation groups of degree n under function composition An, the even permutation groups of degree n under function composition subgroup lattice for s3 http://www.mathhelpforum.com/math-help/advanced-algebra/22850-normal-subgroup.html D3 knot http://www.math.utk.edu/~morwen/3d_pics/more_d3.html reflections of a triangle http://www.answers.com/topic/dihedral-group permutation group A4 http://faculty.smcm.edu/sgoldstine/origami/displaytext.html permutation group s5 http://www.valdostamuseum.org/hamsmith/PDS3.html