Question 3. Letb0 10 01aH =c) : a, b, c e Q1under matrix multiplication. Notice that H is a group (you do not have to prove this).(a) Find Z(H).(b) Prove that Z(H)=Q, Q is under addition.

We will use the basic knowledge of group theory to answer both parts of this question.

Answered: Question 3. Let {G 1 а 0 1 c: a,b, c e…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 2E: 2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .

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True or False Label each of the following statements as either true or false. 10. The nonzero elements of form a group with respect to matrix multiplication.
15. Repeat Exercise with, the multiplicative group of matrices in Exercise of Section. 14. Let be the multiplicative group of matrices in Exercise of Section, let under multiplication, and define by a. Assume that is an epimorphism, and find the elements of. b. Write out the distinct elements of. c. Let be the isomorphism described in the proof of Theorem, and write out the values of.
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
True or False Label each of the following statements as either true or false. 9. The nonzero elements of form a group with respect to matrix multiplication.
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Which of the following subsets in the set R^2×2 is not a group under the usual matrix multiplication operation? Solve (d) and (e)
Let S= \ {-1} and define an operation on S by a*b = a + b + ab. Prove that (S,*) is an abelian group.
(1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group.(2) Find an explicit example of matrices A, B ∈ H(R) such that AB ̸= BA.(3) Is H(R) a subset of GL3 (R)?
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In the given question find an elementary matrix E such that B=EA.

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Question 3. Let {G 1 а 0 1 c: a,b, c e Q 0 0 1 H = %3D under matrix multiplication. Notice that H is a group (you do not have to prove this). (a) Find Z(H). (b) Prove that Z(H)=Q, Q is under addition.