Contemporary Abstract Algebra
Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 2, Problem 6E

In each case, perform the indicated operation.
a. In C*, ( 7 + 5 i ) ( 3 + 2 i )
b. In G L ( 2 , Z 13 ) , det [ 7 4 1 5 ]
c. In G L ( 2 , R ) [ 6 3 8 2 ] 1
d. In G L ( 2 , Z 7 ) [ 2 1 1 3 ] 1

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Question 1. Solve the system - x1 x2 + 3x3 + 2x4 -x1 + x22x3 + x4 2x12x2+7x3+7x4 Question 2. Consider the system = 1 =-2 = 1 3x1 - x2 + ax3 = 1 x1 + 3x2 + 2x3 x12x2+2x3 = -b = 4 1 For what values of a, b will the system be inconsistent? 2 For what values of a, b will the system have only one solution? For what values of a, b will the saystem have infinitely many solutions?
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Chapter 2 Solutions

Contemporary Abstract Algebra

Ch. 2 - Let a belong to a group and a12=e . Express the...Ch. 2 - In U(9)find the inverse of 2, 7, and 8.Ch. 2 - Translate each of the following multiplicative...Ch. 2 - For group elements a, b, and c, express...Ch. 2 - Suppose that a and b belong to a group and...Ch. 2 - Show that the set {5, 15, 25, 35} is a group under...Ch. 2 - Let G be a group and let H=x1xG . Show that G=H as...Ch. 2 - List the members of K=x2xD4andL=xD4x2=e .Ch. 2 - Prove that the set of all 22 matrices with entries...Ch. 2 - For any integer n2 , show that there are at least...Ch. 2 - An abstract algebra teacher intended to give a...Ch. 2 - Let G be a group with the property that for any x,...Ch. 2 - (Law of Exponents for Abelian Groups) Let a and b...Ch. 2 - (SocksShoes Property) Draw an analogy between the...Ch. 2 - Prove that a group G is Abelian if and only if...Ch. 2 - Prove that in a group, (a1)1=a for all a.Ch. 2 - For any elements a and b from a group and any...Ch. 2 - If a1,a2,...,an belong to a group, what is the...Ch. 2 - The integers 5 and 15 are among a collection of 12...Ch. 2 - Prob. 30ECh. 2 - Prob. 31ECh. 2 - Construct a Cayley table for U(12).Ch. 2 - Suppose the table below is a group table. Fill in...Ch. 2 - Prove that in a group, (ab)2=a2b2 if and only if...Ch. 2 - Let a, b, and c be elements of a group. Solve the...Ch. 2 - Let a and b belong to a group G. Find an x in G...Ch. 2 - Let G be a finite group. Show that the number of...Ch. 2 - Give an example of a group with elements a, b, c,...Ch. 2 - Suppose that G is a group with the property that...Ch. 2 - Find an element X in D4 such that R90VXH=D .Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - In the dihedral group Dn , let R=R360/n and let F...Ch. 2 - Prove that the set of all 33 matrices with real...Ch. 2 - Prove that if G is a group with the property that...Ch. 2 - In a finite group, show that the number of...Ch. 2 - List the six elements of GL(2,Z2) . Show that this...Ch. 2 - Prove the assertion made in Example 19 that the...Ch. 2 - Suppose that in the definition of a group G, the...Ch. 2 - Suppose that in the definition of a group G, the...
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