Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
9th Edition
ISBN: 9781305657977
Author: Gallian, Joseph
Publisher: Brooks Cole
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Chapter 2, Problem 10E
Show that the group
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both b anc c if possible
The group SL (2, R) of 2 by 2 matrices with determinant 1 is a normal subgroup of
GL (2, R), the group of 2 by 2 matrices with nonzero determinant. True of false?
O True
O False
Let GL(2,11) be the group of all invertible 2 × 2 matrices with entries in Z₁1,
with group operation given my matrix multiplication. Consider the following two
matrices in this group (where an entry listed as k is shorthand for [k]11):
3
3 10
4- (1₂0). B- (B)
A =
B=
8
8
(ii) Consider the subset of GL(2,11) given by
G= {Am B : m, n = Z}.
Show that G is a subgroup of GL(2, 11).
Chapter 2 Solutions
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following sets are closed under the...Ch. 2 - In each case, find the inverse of the element...Ch. 2 - In each case, perform the indicated operation. a....Ch. 2 - Prob. 7ECh. 2 - List the elements of U(20).Ch. 2 - Show that {1, 2, 3} under multiplication modulo 4...Ch. 2 - Show that the group GL(2,R) of Example 9 is...
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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- True or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.arrow_forwardLet SL₂(R)= {A|A is a 2x 2 matrix with det(A)=1}. We will show that SL₂(R) is a group with matrix multiplication: (a) Let A,BE SL₂(R). Find det(AB). (b) IS AB E SL₂(R) ? O No, SL₂(R) is not closed under multiplication. Yes, SL₂(R) is closed under multiplication. (c) I= We already know that matrix multiplication is associative and that 0 is a multiplicative identity for matrix multiplication. Find det(I). 01 (d) Is IE SL₂(R) ? O No, SL₂(R) does not contain an identity element. O Yes, SL₂(R) contains an identity element. (e) Let AE SL₂(R). Find det(A-¹). (f) Is A-¹ ESL₂(R) ? O Yes, every element of SL2(R) has an inverse in SL₂(R). O No, not every element of SL₂(R) has an inverse in SL₂(R).arrow_forwardShow that G is an abelian under multiplication of matrices ?arrow_forward
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