Consider the multiplicative group GL(2, R), the generallinear group of order 2 over R, and the multiplicativegroup R of non-zero real numbers. Define cp:**a bGL(2,R)->R by p([]) ad bc. Show that p isc dan homomorphism, and find kerp. Is cp anepimorphism? Is op a monomorphism?Consider the multiplicative group GL(2, R), the general linear groupof order 2 over R, and the multiplicative group R* of non-zero realnumbers. Define o GL(2,R) → R* by oa bd• ([ a å ] )= ad -bc. Show that is an homomorphism, and find ker o. Is o anepimorphism? Is ó a monomorphism?

Answered: Image /qna-images/answer/a717a79f-3f62-4792-9bc4-c1cba77fc7b5.jpg

Answered: Consider the multiplicative group GL(2,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 13E: Consider the groups given in Exercise 12. Find an isomorphism from the multiplicative group + of...

See similar textbooks

Similar questions

Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
Consider the additive group and the multiplicative group G={ 1,i,1,i } and define :G by (n)=in. Prove that is a homomorphism and find ker . Is an epimorphism? Is a monomorphism? Find 1(i),1(1),1(i),1(1,1), and 1(i,i).
Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a subgroup of GL(2,R), the general linear group of order 2 over R.The subgroup SL(2,R) is called Special linear group of order 2 over R.
Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.
Let be a homomorphism from the group G to the group G, where G= a , the cyclic group generated by a. Show that is completely determined by the image of the generator a of G.
Let G be a group of finite order n. Prove that an=e for all a in G.
18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
Let G be an abelian group. Prove that the set of all elements of finite order in G forms a subgroup of G. This subgroup is called the torsion subgroup of G.
24. The center of a group is defined as Prove that is a normal subgroup of .
41. Let be a cyclic group, . Prove that is abelian.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

Consider the multiplicative group GL(2, R), the general linear group of order 2 over R, and the multiplicative group R* of non-zero real numbers. Define bc. Show that GL(2,R) R* by o a b ([ å å ]) C d = ad is an homomorphism, and find kero. Is o an epimorphism? Is ó a monomorphism?