1. An isomorphism between groups H and K is a bijection y : H → K that preserves the group operations, that is, writing everything multiplicatively, and writing the action of the bijection on the right, (ab)p = (ap)(bp) for all a, b € H. If this is the case, then we say that H and K are isomorphic, and write H = K . If H and K are groups then the Cartesian product of H and K is H × K = {(a, b) | a E H , b € K} , which becomes a group with coordinatewise group operations (and you do not need to verify this). Throughout this exercise, put G = {: 4.6 € R, a² + b² +0}, c* = {: eC :+ 0}, R+ = {a €R| a > 0} and C = {z € C| |z| =1} . (a) Verify that R+ is a group under multiplication. You may assume any of the usual properties of real numbers. (b) Verify that C is a group under multiplication. You may assume any of the usual properties of complex numbers. (c) Prove that G is an abelian group under matrix multiplication. You may assume any of the usual properties of matrix arithmetic and determinants. (d) Prove that, as multiplicative groups, G = R+ x C = C*. [Hint: use familiar properties of rotation matrices and polar forms of complex numbers.] (It follows that G and C* are isomorphic, because it is easy to check that composites of isomorphisms are isomorphisms.)
1. An isomorphism between groups H and K is a bijection y : H → K that preserves the group operations, that is, writing everything multiplicatively, and writing the action of the bijection on the right, (ab)p = (ap)(bp) for all a, b € H. If this is the case, then we say that H and K are isomorphic, and write H = K . If H and K are groups then the Cartesian product of H and K is H × K = {(a, b) | a E H , b € K} , which becomes a group with coordinatewise group operations (and you do not need to verify this). Throughout this exercise, put G = {: 4.6 € R, a² + b² +0}, c* = {: eC :+ 0}, R+ = {a €R| a > 0} and C = {z € C| |z| =1} . (a) Verify that R+ is a group under multiplication. You may assume any of the usual properties of real numbers. (b) Verify that C is a group under multiplication. You may assume any of the usual properties of complex numbers. (c) Prove that G is an abelian group under matrix multiplication. You may assume any of the usual properties of matrix arithmetic and determinants. (d) Prove that, as multiplicative groups, G = R+ x C = C*. [Hint: use familiar properties of rotation matrices and polar forms of complex numbers.] (It follows that G and C* are isomorphic, because it is easy to check that composites of isomorphisms are isomorphisms.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. An isomorphism between groups H and K is a bijection o : H → K that
preserves the group operations, that is, writing everything multiplicatively, and
writing the action of the bijection on the right,
(ab)y = (ap)(by)
for all a, b e H. If this is the case, then we say that H and K are isomorphic,
and write
H = K.
If H and K are groups then the Cartesian product of H and K is
H x K = {(a,b) | a € H , b € K} ,
%3D
which becomes a group with coordinatewise group operations (and you do not
need to verify this). Throughout this exercise, put
- {; .
a, b eR, a² +b° +0,
C* = {z € C] z +0},
G
a
{a € R|a > 0}
C = {z € C | |z| = 1} .
R+
and
(a) Verify that R+ is a group under multiplication. You may assume any of the
usual properties of real numbers.
(b) Verify that C is a group under multiplication. You may assume any of the
usual properties of complex numbers.
(c) Prove that G is an abelian group under matrix multiplication. You may
assume any of the usual properties of matrix arithmetic and determinants.
(d) Prove that, as multiplicative groups,
G = Rt x C - C*.
[Hint: use familiar properties of rotation matrices and polar forms of complex
numbers.]
(It follows that G and C* are isomorphic, because it is easy to check that
composites of isomorphisms are isomorphisms.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6d236ec-626e-450d-aba0-e87a860b8701%2F57294a9d-a2c2-4e26-9639-58d4f699cb70%2Fxb5phqy_processed.png&w=3840&q=75)
Transcribed Image Text:1. An isomorphism between groups H and K is a bijection o : H → K that
preserves the group operations, that is, writing everything multiplicatively, and
writing the action of the bijection on the right,
(ab)y = (ap)(by)
for all a, b e H. If this is the case, then we say that H and K are isomorphic,
and write
H = K.
If H and K are groups then the Cartesian product of H and K is
H x K = {(a,b) | a € H , b € K} ,
%3D
which becomes a group with coordinatewise group operations (and you do not
need to verify this). Throughout this exercise, put
- {; .
a, b eR, a² +b° +0,
C* = {z € C] z +0},
G
a
{a € R|a > 0}
C = {z € C | |z| = 1} .
R+
and
(a) Verify that R+ is a group under multiplication. You may assume any of the
usual properties of real numbers.
(b) Verify that C is a group under multiplication. You may assume any of the
usual properties of complex numbers.
(c) Prove that G is an abelian group under matrix multiplication. You may
assume any of the usual properties of matrix arithmetic and determinants.
(d) Prove that, as multiplicative groups,
G = Rt x C - C*.
[Hint: use familiar properties of rotation matrices and polar forms of complex
numbers.]
(It follows that G and C* are isomorphic, because it is easy to check that
composites of isomorphisms are isomorphisms.)
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