301 Abstract Algebra 2:00 - 2:50 MWF Hentzel 432 Carver hentzel@iastate.edu Web Page http://www.math.iastate.edu/hentzel/class.301 Text: A First Course in Abstract Algebra, sixth Edition by John B. Fraleigh Course: Chapter 1,2,3,4,5 Goals: Sylow theorems Burnside counting theorem Fundamental Theorem of Finite Abelian Groups Polynomials Start: The material we are covering today is on Page 108-113 section 2.2 beginning with Cycles. I give you this reference with hesitation because I do not want you to become discouraged trying to jump into a book in the middle of the second chapter. The presentation there is a bit obtuse because it is given with terms you are not familiar with. Instead, pay attention to the class notes. These will be self contained. We will eventually go back and cover the initial sections. Permutations on three objects: (I REWROTE THESE USING RIGHT TO LEFT NOTATION) ------------------------------ phi * * * "first two" <---> psi ---->----> "cycle" * * * <--------- 2 ----------> psi * * * <----<----- ----------> psi phi * * * "outside" <--------- phi psi * * * "two three" <----> I * * * "identity" 3 psi = I "psi cubed is the identity" 2 phi = I "phi squared is the identity" 2 phi psi = psi phi "bringing phi across psi changes the exponent" 2 psi phi = phi psi Simplify the following product: 2 phi psi phi psi psi phi phi psi phi I phi phi psi A man has a duck, a fox, and a bag of corn. The fox will eat the duck. The duck will eat the bag of corn. The man comes to a stream. He only wants to carry one item across at a time. He dare not leave the fox and duck together, or the duck and the bag of corn together. How does he cross the stream. He takes the duck across first. Then he returns and grabs the bag of corn. Halfway across he realizes he has a problem. | | fox | bag | duck | | He can go back and take the fox which means he can do phi. He can go forth and bring back the duck which is psi phi. He realizes the allowable acts are { phi, phi psi}. He knows that he before he got into this fix he had all the animals on the left bank. If he reverses things getting | | duck | bag | fox | | he can land the bag of corn and the duck and will have successfully crossed the stream. Therefore, he must get psi phi using only phi and phi psi phi (phi psi) phi = psi phi. first do phi, then (phi psi), then do phi. (phi) = (12) psi = (123) psi^2 = (132) phi psi = (12)(123) = (1)(23) psi phi = (123)(12) = (13)(2) psi^2 phi = (132)(12) = (1)(23) phi psi^2 = (12)(132) = (13)(2) and so phi psi = (1)(23) = psi^2 phi and psi phi = (13)(2) = phi psi^2 There is a better way to write out permutations than by giving them names like phi and psi. This better way is called cycle notation. a b c d e f f c d a b e One numbers the positions: 1 2 3 4 5 6 a b c d e f f c d a b e This is the permutation given in cycle notation. (1,4,3,2,5,6) Any cyclic shift represents the same permutation. (4,3,2,5,6,1) or (3,2,5,6,1,4) or (2,5,6,1,4,3) or ? or ? More often than not, we start the cycle with the smallest element. This makes it easier to construct the cycles. 1 2 3 4 5 6 a b c d e f b a f d c e (1,2)(3,5,6)(4) This is the cycle notation. Multiplication in cyclic notation: (1,2,3)(4,5) (3,5,2,1,4) = (1,5,3,4)(2) Notice that Thus (12) = phi (123) = psi 2 (132) = psi and (12)(123) = (1)(23) (123)(12) = (13)(2) 2 psi = (132) ###################################################### FROM NOW ON WE WILL ALWAYS MULTIPLY FROM RIGHT TO LEFT. Just like the text book. Any you can forget everything you learned about {I,phi,psi,phi psi, psi phi, psi^2} ###################################################### Some observations: ------------------ n (a) If Pi a cycle of length n, then Pi = I. (b) If Pi is a product of disjoint cycles of lengths n l1, l2, ..., lr, then Pi = I where n = LCM(l1,l2, ..., lr). (c) Disjoint Cycles commutes. Cycles which are not disjoint may not commute. (d) The inverse of a cycle is obtained by writing the cycle backwards. The inverse of a product of cycles in obtained by writing the cycles and everything inside the cycles backwards. Example: -------- T O M B Pi W H O M H U G [] ==========> B I G [] W I R E D U S T D A Y S Y E A R Use the numbering system 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Pi = ( 1,12,14,15,13, 9)( 2, 3, 4, 5)( 6,10)( 7)( 8),(11,16) 2 Pi = ( 1,14,13)(12,14,9)( 2, 4)( 3, 5)( 6)(10)( 7)( 8)(11)(16) LCM(6,4,2,1,1,2) = 12, so the twelfth power of Pi is I. LCM(3,3,2,2,1,1,1,1,1,1) = 6, so the sixth power of Pi^2 is I. -1 Pi = (16,11)( 8)( 7)(10,6)( 5, 4, 3, 2)( 9,13,15,14,12, 1) ##################################################################### Assignment: Due Wednesday, August 29. ? ? ? ? -1 B A Y [] G U M [] ? ? ? ? ? ? ? ? <--Pi---- G U S H ----Pi--> B I D S ---Pi--> ? ? ? ? ? ? ? ? W O R M W H A T ? ? ? ? ? ? ? ? T I D E Y O R E ? ? ? ? (1) Write the orbit notation for Pi. 2 B A Y [] (2) Find Pi of G U S H W O R M T I D E -1 B A Y [] (3) Find Pi of G U S H W O R M T I D E (4) Write the orbit notation of a perfect shuffle. (5) Write the symmetry group of a square.