(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)? Exercise 3.2.13. LetH(R)a с={(1 + $):ARCER}01 b a, b, c001(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)?

The main objective is to prove H(R) is group.

Exercise 3.2.13. Let H(R) a - {(1; i)=ancex} = a, b, 0 0 1 (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)?

Exercise 3.2.13. Let H(R) a - {(1; i)=ancex} = a, b, 0 0 1 (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)?

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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Question

(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)?

Exercise 3.2.13. Let
H(R)
a с
=
{(1 + $):ARCER}
01 b a, b, c
001
(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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