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# MODERN ALGEBRA I MATH 225

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This 2 page Class Notes was uploaded by Donald Gusikowski on Monday October 5, 2015. The Class Notes belongs to MATH 225 at Clark University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/219539/math-225-clark-university in Mathematics (M) at Clark University.

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Date Created: 10/05/15

Math 225 Modern Algebra Midterm 14 Oct 2003 You may refer to the one sheet of notes that you prepared7 but nothing else Write your answers in a bluebook You may do the problems in any order that you like just be sure to start each problem on a new page of the bluebook Problem 1 On permutations and their matrix representations 20 Recall that the symmetric group Sg can be described as the set 53 l cwzwwy czy where x3 17y2 1 and ym y a Exhibit a 3 gtlt 3 matrix that represents the permutation z and another 3 gtlt 3 matrix that represents y b Using your representation verify that ym y in 3 c Show that the permutation y is an odd permutation while z is an even permutation by evaluating their signs 1 ls the order of y even or odd Problem 2 On basic properties of groups 20 Recall the de nition of group that we7ve used in this course A group is a set G together with a binary operation that satis es the following three axioms 1 Associatvity Vxyz 2 Identity 3x Vy xy y yx Such an x is called an identity element 3 lnverses Vx 3y xy ym 17 where 1 is an identity element as described in axiom a Note that axiom 2 says that there is at least one identity element Prove that the identity element is unique Use only the three axioms for groups mentioned in the de ntion7 and point out every time you use an axiom in your proof b Prove the cancellation law for groups if my and z are elements in a group G then x2 yz implies z y Use only the three axioms for groups mentioned in the de ntion and the results of part a which allow you to denote the unique group identity as 1 in your proof Point out every time you use an axiom or part a in your proof Problem 3 On the quaternion group 30 You dont have to prove your assertions in this problem Recall that one way to describe the quaternion group H is that it contains the eight elements i1 i 7ij7ik and these have the properties 2392 j2 k2 712739 kjk mz39 j In a homework problem you worked out a multiplication table for H 171 2394 jij kik 1171 2394 jij kik 7 7 ij j k 7k 1 i1 7239 239 k k 7k j 7 7 7239 239 i1 1 7k 7k k 7j j 239 7239 1 i1 a ls H an Abelian group b ls H a cyclic group c Which of the 8 elements is the conjugate z ji l 1 Name at least one subgroup of H of order 2 ls it normal or not e List all the left cosets of the subgroup that you just named f Name a subgroup of order 4 g ls the subgroup you just mentioned a normal subgroup h Are there any quotient groups of H of order 2 i Are there any quotient groups of order 4 j ls H the product of smaller groups Problem 4 On kernels of group homomorphisms 20 Let f G a H be a group homomorphism7 and let K ker f x E Gfx 1 be the kernel of f a Prove that K is a subgroup of G b Prove that K is a normal subgroup of G Problem 5 On modular arithmetic 15 You dont have to prove your assertions in this problem The set of congruence classes modulo a xed integer n is usually denoted ZnZ It is written additively7 and its an Abelian group a How many generators of the cyclic group Z10Z are there Name them b How many subgroups of Z10Z are there Name them You can list their elements if you like c What are the possible orders of the elements of Z30Z Of course7 1 and 30 are two orders that occur What are the others

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