Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 461 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://architecturebuildingconstruction.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=12686 870 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. Caution: In Matrix Multiplication, (A*B)*C=A*(B*C). The Commutative Property, also familiar from ordinary arithmetic on real numbers, does not generally apply to all groups! In function composition, f*(g*h) = (f*g)*h. Only Abelian groups are commutative. This may take some “getting-used-to,” at first! This is a property of all groups. 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. |1 0| = I |0 1| The Identity is Unique! There is only one identity element in any group. This property is used in proofs. 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. 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 nsided 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 permutation group A4 http://faculty.smcm.edu/sgoldstine/origami/displaytext.html permutation group s5 http://www.valdostamuseum.org/hamsmith/PDS3.html D3 knot http://www.math.utk.edu/ ~morwen/3d_pics/more_ d3.html reflections of a triangle http://www.answers.com/topic/di hedral-group subgroup lattice for s3 http://www.mathhelpfor um.com/mathhelp/advancedalgebra/22850-normalsubgroup.html Finite Groups and Subgroups, Terminology At this time we are mainly concerned with finite groups, that is, groups with a finite number of elements. The order of a group, |G|, is the number of elements in the group. The order of a group may be finite or infinite. The order of an element, |a|, is the smallest positive integer n such that an = e. The order of an element may likewise be finite or infinite. Note: if |a|=2 then a=a-1. If |a|=1 then a=e. A subgroup H of a group G is a subset of G together with the group operation, such that H is also a group. That is, H is closed under the operation, and includes inverses and identity. (Note: H must use the same group operation as G. So Zn, the integers mod n, is not a subgroup of Z, the integers, because the group operation is different.) euler portrait http://www.math.o hiostate.edu/~sinnott/ ReadingClassics/h omepage.html Cancellation and Conjugation In any group, a*b=a*c implies that b=c and c*a=b*a implies that c=b. This is used in proofs. To conjugate an element a by x means to multiply thus: xax-1 or x-1ax While conjugating an element may change its value, the order |a| is preserved. This is useful in proofs and in solving matrix equations. cancellation and conjugation http://keelynet.com/indexfeb206.htm “Socks and Shoes” Property When taking inverses of two or more elements composed together, the positions of the elements reverse. That is, (a*b)-1 = b-1*a-1. For more elements, this generalizes to (ab...yz)-1 = z-1y-1...b-1a-1. In Abelian groups, it is also true that (ab)-1 = a-1b-1 and (ab)n = anbn. This also generalizes to more elements. This is called the “socks and shoes property” as a mnemonic, because the inverse of putting on one's socks and shoes, in that order, is removing ones shoes and socks, in that order. shoes and socks in the car http://picasaweb.google.com/mp3873/PAD#5235584405081556594 shoes and socks http://www.inkfinger.us/my_weblog/2007/04/index.html Subgroup Tests: The One Step Subgroup Test Suppose G is a group and H is a non-empty subset of G. If, whenever a and b are in H, ab-1 is also in H, then H is a subgroup of G. Or, in additive notation: If, whenever a and b are in H, a - b is also in H, then H is a subgroup of G. -1 ab H Example: Show that the even integers are a subgroup of the Integers. Note that the even integers is not an empty set because 2 is an even integer. Let a and b be even integers. Then a = 2j and b = 2k for some integers j and k. a + (-b) = 2j + 2(-k) = 2(j-k) = an even integer Thus a - b is an even integer Thus the even integers are a subgroup of the integers. To apply this test: Note that H is a nonempty subset of G. Show that for any two elements a and b in H, a*b-1 is also in H. Conclude that H is a subgroup of G. one step at a time by norby http://www.flickr.com/photos/norby/37932 1413/ Subgroup Tests: The Two Step Subgroup Test Let G be a group and H a nonempty subset of G. If a●b is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G. Example: show that 3Q*, the non-zero multiples of 3n where n is an integer, is a subgroup of Q*, the non-zero rational numbers. To Apply the Two Step Subgroup Test: Note that H is nonempty Show that H is closed with respect to the group operation Show that H is closed with respect to inverses. Conclude that H is a subgroup of G. 3Q* is non-empty because 3 is an element of 3Q*. For a, b in 3Q*, a=3i and b=3j where i, j are in Q*. Then ab=3i3j=3(3ij), an element of 3Q* (closed) For a in 3Q*, a=3j for j an element in Q*. Then a-1=(j-1*3-1), an element of 3Q*. (inverses) Therefore 3Q* is a subgroup of Q*. http://www.trekearth.com/gallery/Asia/Brunei/photo653317.htm Subgroup Tests: The Finite Subgroup Test Let H be a nonempty finite subset of G. If H is closed under the group operation, then H is a subgroup of G. To Use the Finite Subgroup Test: If we know that H is finite and non-empty, all we need to do is show that H is closed under the group operation. Then we may conclude that H is a subgroup of G. Example: To Show that, in Dn, the rotations form a subgroup of Dn: Note that the set of rotations is non-empty because R0 is a rotation. Note that the composition of two rotations is always a rotation. Therefore, the rotations in Dn are a subgroup of Dn. math cartoons from http://www.math.kent.edu/~sather/ugcolloq.html Examples of Subgroups: cyclic subgroups Let G be a group, and a an element of G. Let <a> = {an , where n is an integer}, (that is, all powers of a.) ...Or, in additive notation,,, let <a>={na, where n is an integer}, (that is, all multiples of a.) Then <a> is a subgroup of G. Note: In multiplicative notation, a0 = 1 is the identity; while 0a=0 is the identity in additive notation. Thus <a> includes the identity. For example: In R*, <2>, the powers of 2, form a subgroup of R*. In Z, <2>, the even numbers, form a subgroup. In Z8, the integers mod 8, <2>={2,4,6,0} is a subgroup of Z8. In D3, the dihedral group of order 6, <R120> = {R0, R120, R240} is a subgroup of D3 Also note that the integers less than 0 are included here, so <a> includes all inverses. Each element generates its own cyclic subgroup. subgroup image http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png Examples of Subgroups: The Center of a Group Z(G) The Center of a group, written Z(G), is the subset of elements in G which commute with all elements of G. If G is Abelian, then Z(G)=G. If G is non-Abelian, then Z(G) may consist only of the identity, or it may have other elements as well. For example, Z(D4) = {R0, R180}. The Center of a Group is a Subgroup of that group. Subgroup lattice for D3 http://mathworld.wolfram.com/DihedralGroupD3.html Examples of Subgroups: The Centralizer of an Element C(a) The Centralizer of an element C(a): For any element a in G, the Centralizer of a, written C(a) is the set of all elements of G which commute with a. In an Abelian group, C(a) is the entire group. In a non-Abelian group, C(a) may consist only of the identity, a, and a-1, or it may include other elements as well. For example, in D3, C(f) ={f, R0}, while C(R0)=D3 For each element a in a group G, C(a) = the centralizer of a is a subgroup of G. subgroup image http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png Cyclic Groups A Cyclic Group is a group which can be generated by one of its elements. That is, for some a in G, G={an | n is an element of Z} Or, in addition notation, G={na |n is an element of Z} This element a (which need not be unique) is called a generator of G. Alternatively, we may write G=<a>. http://www.math.csusb.edu/faculty/susan/modular/modular.html Examples: (Z,.+) is generated by 1 or -1. Zn, the integers mod n under modular addition, is generated by 1 or by any element k in Zn which is relatively prime to n. Non-Examples: Q* is not a cyclic group, although it contains an infinite number of cyclic subgroups. U(8) is not a cyclic group. Dn is not a cyclic group although it contains a cyclic subgroup <R(360/n)> Properties of Cyclic Groups: Criterion for For |a| = n, ai = aj iff n divides (i-j) (alternatively, if i=j mod n.) Or, in additive notation, ia = ja iff i=j mod n. Corollaries: 1. |a|=|<a>| that is, the order of an element is equal to the order of the cyclic group generated by that element. 2. If ak=e then the order of a divides k. math cartoons from http://www.math.kent.edu/~sather/ugcolloq.html i a = j a For example, in Z5, 2x4 = 7x4 = 3 because 2=7 mod 5 For example... in Z10, |2|=5 and <2>={2,4,6,8,0} Caution: This is why it is an error to say, that the order of an element is the power that you need to raise the element to, to get e. A correct statement is, the order of an element is the smallest positive power you need to raise the element to, to get e. Properties of Cyclic Groups k gcd(n,k) <a >=<a > For |a|=n, n k and |a |= /gcd(n,k) In words, this reads as: If the order of a is n, then the cyclic group generated by a to the k power is the same as the cyclic group generated by a to the power of the greatest common divisor of n and k. Also, the order of a to the k power is equal to the order of a divided by the greatest common divisor of k and the order of a. (Exercise: Try verbalizing a similar statement for additive notation!) For example ... In Z30, let a=1. Then |a|=30. Since Z30 uses modular addition, a26 is the same as 26. What is the order of 26? Since gcd (26,30)=2, and gcd(2,30)=2, it follows that |26|= |2| = gcd(2,30)=15. Thus we expect that <26>=<2>, and, in fact, this is {0,2,4,6,8...24,26,28}. So we see that |<2>|=|<26>| =30/gcd(2,30) = 30/2 = 15. “You may have to do something like this in a stressful situation” - Dr. Englund Properties of Cyclic Groups k gcd(n,k) <a >=<a > For |a|=n, and |ak|=n/gcd(n,k) Corollary 1: When are cyclic subgroups equal to one another? Let |a|=n. Then <ai>=<aj> iff gcd(n,i)=gcd(n,j) This gives us an easy way to specify the generators of a group, the generators of its subgroups, and to tell how these are related. Corollaries.... For example ... In Z30, let a=1. Again, consider a2 and a26. Since gcd (26,30)=2, and gcd(2,30)=2, it follows that <|26|>= <|2|> = gcd(2,30)=15. Thus we expect that <26>=<2>, and, in fact, this is {0,2,4,6,8...24,26,28}. So we see that |<2>|=|<26>| =30/gcd(2,30) = 30/2 = 15. On the other hand, <3> ≠<2> because gcd(30,3) = 3 while gcd (30,2)=2. And in fact, <3>={0,3,6...24,27} and |<3>|=30/3 = 10. However, <3>=<9>. Do you see why? Properties of Cyclic Groups k gcd(n,k) <a >=<a > For |a|=n, and |ak|=n/gcd(n,k) Corollary 2: Generators of Cyclic Groups In any cyclic group G=<a> with order n, the generators are ak for each k relatively prime to n. This gives an easy way to find all of the generators of a cyclic group. Corollaries.... For example ... In Z10, let a=1, so that Z10 = <a>. Other generators for Z10 are ak for each k less than 10 and relatively prime to 10. So the other generators are 3,7,and 9. Corollary 3 specifies this for Zn , the integers mod n under modular addition. gauss stamp Since any Zn is a cyclic group of order n, http://webpages.math.luc.e du/~ajs/courses/322spring its generators would be the positive integers less than n 2004/worksheets/ws5.html and relatively prime to n. Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups Let G=<a> be a cyclic group of order n. Then ... 1. Every subgroup of a cyclic group is also cyclic. 2. The order of each subgroup divides the order of the group. 3. For each divisor k of n, there is exactly one subgroup of order k, that is, <a n/k > symmetry 6 ceiling art http://architecture-buildingconstruction.blogspot.com/2006_03_01_archive.html For example consider Z10 = <1> with |1|=10. Let a=1. Every subgroup of Z10 is also cyclic. The divisors of 10 are 1, 2, 5, and 10. For each of these divisors we have exactly one subgroup of Z10, that is, <1>, the group itself, with order 10/1=10 <2>={0,2,4,6,8} with order 10/2 = 5 <5>={0,5} with order 10/5 = 2 <10>={0} with order 10/10=1 The order of each of these subgroups is a divisor of the order of the group, 10. So the generators of Z10 would be 1, and the remaining elements: 3, 7, and 9. Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups - Corollary Corollary -Subgroups of Zn: For each positive divisor k of n, the set <n/k> is the unique subgroup of Zn of order k. These are the only subgroups of Zn. For example consider Z10 = <1> with |1|=10. Let a=1. Every subgroup of Z10 is also cyclic. The divisors of 10 are 1, 2, 5, and 10. For each of these divisors we have exactly one subgroup of Z10, that is, <1>, the group itself, with order 10/1=10 <2>={0,2,4,6,8} with order 10/2 = 5 <5>={0,5} with order 10/5 = 2 <10>={0} with order 10/10=1 The order of each of these subgroups is a divisor of the order of the group, 10. So the generators of Z10 would be 1, and the remaining elements: 3, 7, and 9.. Number of Elements of Each Order in a Cyclic Group Let G be a cyclic group of order n. Then, if d is a positive divisor of n, then the number of elements of order d is φ(d) where φ is the Euler Phi function φ(d) is defined as the number of positive integers less than d and relatively prime to d. The first few values of φ(d) are: d 1 2 3 4 5 6 7 8 9 10 11 12 φ(d) 1 1 2 2 4 2 6 4 6 4 10 4 In non-cyclic groups, if d is a divisor of the order of the group, then the number of elements of order d is a multiple of φ(d), euler totient graph http://www.123exp-math.com/t/01704079357/ For example, consider Z12 ... Z12 = {0,1,2,3,4,5,6,7,8,9,10,11} We have 1 element of order 2 = {6} because φ(2)=1. We have 2 elements of order 3 = {4,8} because φ(3)=2. We have 2 elements of order 4 = {3,9} because φ(4)=2. And 2 elements of order 6 = {2,10} because φ(6)=2 euler totient equation http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80 Permutation Groups – Definition Definitions A Permutation of a set A is a function from A to A which is both one to one and onto So in a Permutation on set A, the range and the domain of the function are both the set A. Recall from earlier work: A function from set A to set B is a rule which assigns to each element of A (the domain) exactly one element of B (the range) One to one means f(a)=f(b) implies that a=b. Onto means every element of B is the image of least one element of A under f. rubic cube permutation group http://en.wikipedia.org/wiki/Permutation_group Permutation Groups – Definitions Definitions: A Permutation Group of a set A is a set of Permutations on A which form a group under function composition. That is, the elements are functions from A to A which are both one to one and onto, and the binary operation is function composition, Recall from earlier work: A group has elements which include identity and inverses, and an operation which is associative and closed. So here, the elements are functions and the operation is function composition, which is always associative (although not usually commutative.) This may take some “getting used to”, because in most of the groups we have seen so far, the elements are values and the operation is a function. Permutation Groups -- Discussion So in a Permutation Group, we need functions and their inverses; also we need an identity function. Recall from earlier work that a function has an inverse if and only if it is one to one and onto. Since functions are group elements, we expect that a function composed with its inverse will result in the identity function, as indeed it does. At this time we are mainly concerned with permutations on a finite set, so we can build closure into the function definitions. Permutation Groups – Notations Notations In other areas such as algebra and calculus, functions are defined on infinite sets and often written as algebraic formulas. However, in this context we usually define a function explicitly, by listing an element of the domain along side its corresponding function value. There are two ways this is commonly done: grid notation (due to Euler) and cycle notation (due to Cauchy). math cartoons from http://www.math.kent.edu/~sather/ugcolloq.html Permutation Groups – Notations Example, D4, the symmetries of a Square Consider D4, the symmetry group of a square. We can represent this group as a permutation group in grid notation as follows. Number the four corners of the square: 1,2,3,4 as shown. Or, in cycle notation we can write: R90 = (1 2 3 4) and Fh= (1 2)(3 4) 3 2 4 1 Then we can represent a 90 o counterclockwise rotation as: R90 = 1 2 3 4 2341 and a reflection across the horizontal axis as: Fh = 1 2 3 4 2143 Permutation Groups – Definitions and Notation S3 and Sn Let S3 be the group of all one to one functions from the set {1, 2, 3} to itself. This is the same as saying, all arrangements or permutations of these three elements. Recall from previous work that the number of permutations on a set of n elements is n! So the number of elements in S3 is 3! or 6. In general, Sn is the set of all permutations on a set of size n, and the order of Sn is n! subgroup lattice for s3 http://www.mathhelpforum.com/math-help/advanced-algebra/22850-normalsubgroup.html Cauchy Cycle Notation Disjoint Cycles Commute Order of a Permutation Cycle notation, introduced by Cauchy, has many advantages over grid notation. In general, the product of cycles does not commute. However, we can write any permutation as a cycle or a product of disjoint cycles, that is, each element of the set appears in at most one cycle. Written in this form, disjoint cycles commute. Also when we write a permutation in this form, the order of the permutation (that is, the smallest number of times we need to repeat that permutation to obtain the identity permutation) is the least common multiple of the lengths of the cycles. Cycle Notation Decomposition into two-cycles Always even or always odd Identity is always even Any permutation may also be written or “decomposed” as a product of two-cycles. This product would usually not be disjoint, and it need not be unique. However, if a permutation can be written as an odd number of two-cycles, any other decomposition of that permutation is also an odd number of two cycles. Similarly, if a permutation can be written as an even number of two-cycles, any other decomposition of that permutation is also an even number of two cycles. The identity permutation is always an even number of two-cycles. “I’m odd and I’ll always be odd!-- Joni This oddfellow is still odd even though he’s decomposed! IOOF tombstone http://farm1.static.flickr.com/203/485879582_fa65f4e4b1.jpg?v=0 Cycle Notation Definitions, Examples Even Permutation Odd Permutation If a permutation can be written as an odd number of two cycles, we say it is an odd permutation. Examples of decomposition into 2cycles: (12345) = (15)(14)(13)(12) (1234) = (14)(13)(12) (1234)(247)=(14)(13)(12)(27)(24) If a permutation can be written as an even number of two cycles, we say it is an even permutation. “A cycle of odd length is always even – now that's odd! “ -- Rus Notice that (12345) is a cycle of length 5, and 5 is an odd number – yet (12345) can be written as an even number of two-cycles so it is an even permutation. However, (1234), a cycle of length 4 is an odd permutation because it can be written as an odd number of two-cycles, even though 4 is an even number. An The Alternating Groups of Degree n For any n, the set of even permutations in Sn forms a group, and a subgroup of Sn. This is because the inverse of an even permutation is always even, the composition of even cycles is always even, and the identity can always be written as an even cycle. This group is called An, the alternating group of degree n. When n ≠1, An is of order n!/2 Permutation group A5 http://www.math.metu.edu.tr/~berkman/466object.html The Alternating groups are very important in historical, theoretical, and applications contexts. permutation group A4 http://faculty.smcm.edu/sgoldstine/origami/display text.html Note that the set of odd permutations do not form a group, because this set lacks the identity permutation and lacks closure under composition. Suggested Activities Cycle Notation, more practice Cycle notation, introduced by Cauchy, has many advantages over Euler's grid notation, although some people may require more practice to perform calculations quickly and accurately in this notation. For practice, try verifying the Cauchy table for A4 on page 105 of the text, by multiplying out each pair of elements and showing that the product is as stated in the table. Suggested Activities Reinforcement of Knowledge Base Modern educational practices tend to de-emphasize knowledge base in favor of constructivist education theory, which is thought to better support higher order thinking. However, for most people*, an extensive knowledge base is necessary for mastery of this subject. Please see the textbook author Gallian’s comments this matter, on his website http://www.d.umn.edu/~jgallian under the heading “Advice for students for learning abstract algebra” *Sylvester notwithstanding, text pg 89 Tools 3 www.istockphoto.com tools and rocks www.dkimages.com One who has the facts memorized with understanding is comparable to a highly skilled worker who has the proper tools on hand and knows how to use them, rather than having to scrounge around and improvise with a stick or a rock! Suggested Activities: Reinforcement of Knowledge Base according to individual learning style Often students complain that memorizing facts and definitions is excessively difficult, or that knowledge obtained this way does not seem to apply to testing or problem solving situations. Complaints like this often result from failure to adapt study habits to one's particular learning style. For example, a visual learner may get by with simply reading over text or notes, while an aural learner may need to say lessons out loud or even set them to music. A kinesthetic learner often needs to re-write the lessons, make models, demonstrate the information to others, etc. Although kinesthetic methods are more demanding in terms of time and energy, all learners benefit from kinesthetic learning methods, because these methods produce more robust memory traces in the brain. Suggested Activities: Reinforcement of Knowledge Base according to individual learning style Please visit one or both of these sites to find out or confirm your learning style: http://www.usd.edu/trio/tut/ts/stylestest.html for a brief quiz and classification into one of three styles, or http://www.learning-styles-online.com for a more extensive questionnaire and classification into one of six styles. This site will generate a detailed description of your learning style and a graph showing your style according to six axes. Both sites have been checked for safety from malware, and contain recommendations and resources for the various learning styles. This is a sample Memletics Learning Styles graph produced by the online test at http://www.learning-styles-online.com/inventory/ Picture credit http://www.learning-styles-online.com/inventory/ Suggested Activities: Online Resources provided by Text Author J.A. Gallian http://www.d.umn.edu/~jgallian Practice with the true-false questions and flashcards provided on this website may be particularly helpful for building and reinforcing knowledge base. The flashcards may be printed out or used in software form, from the website. This website also contains links to some useful software. In particular, the Group Explorer software, available on the above website by following the link Group Explorer, or one may go directly to: http://groupexplorer.sourceforge.net/ quarternion group from http://home.att.net/~numericana/ answer/groups.htm rock stack from loveringllc.com Suggested Activities: Other Online Resources Here are a few other good online resources. All of these have been checked for malware and are safe. http://en.wikipedia.org/wiki/Group_theory (a good starting point) http://www.math.miami.edu/~ec/book/ (a free online book on abstract algebra) http://en.wikipedia.org/wiki/Dihedral_group (explains the dihedral groups) http://members.tripod.com/~dogschool/cyclic.html I (explains Cauchy cycle notation) http://en.wikipedia.org/wiki/Euler%27s_totient_function (Totient function, in Gallian text referenced in connection with U-groups) http://members.tripod.com/~dogschool/index.html (an index to another online text) http://en.wikipedia.org/wiki/Symmetric_group (explanations and applications of symmetric groups) http://www.t209.com/article.php?art_id=26 (a vbasic program φ(n) for numbers up to 231) http://eldar.mathstat.uoguelph.ca/dashlock/math3130/pdf/Chapter2b.pdf (another explanation of Cayley cycle notation) http://www.mathwire.com/seasonal/winter07.html Acknowledgments Dr. T. Englund, Professor of Mathematics, Central Washington University, professor for Mathematics 461 Dr. T. Willard, Professor of Mathematics, Central Washington University, graduate adviser I am grateful for the cooperation of the following fourth year undergraduate Mathematics students at Central Washington University, who included me in their study group for the final exam. Notes from this study group were then composed into this presentation: Amber Goodrich, Mike Prothman, Russel Hess, David Melik, Geoff LaBrandt, Brandon Belieu, Special thanks to: Two Bears, my brother St. Euler St. Cauchy All graphics in this presentation not created by J. Pinkney are reproduced from online sources according to the Fair Use Provisions of the U.S. Copyright Act, with the html sources cited in the frame on which they appear.