<p>MTH 200 – Graded Assignment 9 </p><p>Material from Module 12 (Groups – sections 4.1, 4.2, 4.4).</p><p>P1: What are the operator properties that must be satisfied for a group? What is the additional property required for an Abelian group?</p><p>P2: </p><p> a) Write an operation table for U14 . </p><p> b) What is the value of 11-1 ?</p><p> c) In U14 , solve 5x = 9 . P3: What is the order of the group U315 ?</p><p>P4: In the Klein 4-group a) What is the value of (1,1)-1 ?</p><p> b) Solve the equation (1,0) �X (0,1) .</p><p> c) Prove that the operation as defined for the Klein 4-group is associative (you should be able to get this directly from the definition, without having to considers all possible combinations of elements. n n P5: Show that in U8 , a = a if n is odd, and a =1if n is even.</p><p>P6: Prove that if <G,* > is a finite group and x G , then there is a positive integer n such that xn = e (the identity). P7: Prove the closure of Um for any m ; i.e. prove that the product of two units in Zm under multiplication modulo m is also a unit in Zm . I’d suggest indirect or contradiction; suppose that a and b are units in Zm , but the product ab is not, so that gcd(ab , m )= d > 1 , and take it from there.</p><p>P8: In the dihedral group of the triangle D3</p><p> a) Compute Fb * R2 . Describe how the triangle has been transformed.</p><p> b) Solve the equation Fc* X= F a , explicitly showing the use of inverses.</p><p> c) For which elements X D3 does R2*X = X * R 2 (which elements does R2 commute with)? P9: In the dihedral group of the square D4</p><p> a) Compute R1* D 1 . Describe how the square has been transformed.</p><p> b) Solve the equation V* X= D1 , explicitly showing the use of inverses.</p><p> c) Solve the equation X* R3= R 2 , explicitly showing the use of inverses.</p><p>P10: What would be the order of the group of symmetries of a regular octagon? Through what angles can the octagon be rotated that will map it congruently back onto itself? P11: Find each of the following cyclic groups (give the set that the group generates).</p><p> a) <7 > in U16</p><p> b) <4 > in Z10</p><p> c) <(1,1) > in the Klein 4-group</p><p>P12: In <, + > a) Find <5,8 ></p><p> b) Find <6,15 > P13: Consider Z12 , and let H= { mx | x Z12 }. Construct the subgroup H for a) m = 3</p><p> b) m = 5 P14: Take the dihedral group D4 and find the cyclic subgroup generated by each of the elements (go though each in turn, I’ve done H as an example). Give the order of each element in D4 . Is D4 itself a cyclic group?</p><p>< R0 > : <H > : (H)1 = H 2 (H ) =H * H = R0</p><p>< H >= {R0 ,H } | H |= 2</p><p>< R1 > : <V > :</p><p>< R2 > : <D1 > :</p><p>< R3 > : <D2 > : P15: In U14 , find a) <9 ></p><p> b) | 9 |</p><p> c) <5,11 ></p><p> d) An element x U14 such that <x >=<11 > . You can answer this without even working out what <11 > is.</p><p> e) Is U14 cyclic? If yes, give a generator. P16: Consider the direct product group U14 Z 3 a) List all the elements in the group.</p><p> b) How is the *operation defined for elements of U14 Z 3 ?</p><p> c) Compute (11,2)*(3,2) .</p><p> d) Find the inverse of (11,1) .</p><p> e) Use the inverse to solve the equation (11,1)*X = (5,2) .</p><p> f) Find a proper subgroup of U14 Z 3 by crossing subgroups of U14 and Z3 (at least one of those must be proper in order for the cross to be proper).</p>