College Algebra Exam Review 136

College Algebra Exam Review 136 - morphism is completely...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
146 2. BASIC THEORY OF GROUPS 2.7.5. Consider the set of fractional linear transformations of the complex plane with 1 adjoined, C [ f1g , T a;b I c;d .z/ D az C b cz C d where a b c d is an invertible 2-by-2 complex matrix. Show that this is a group of transformations and is isomorphic to GL .2; C /=Z. GL .2; C //: 2.7.6. Recall that an automorphism of a group G is a group isomorphism from G to G . Denote the set of all automorphisms of G by Aut .G/ . (a) Show that Aut .G/ of G is also a group. (b) Recall that for each g 2 G , the map c g W G ! G defined by c g .x/ D gxg 1 is an element of Aut .G/ . Show that the map c W g 7! c g is a homomorphism from G to Aut .G/ . (c) Show that the kernel of the map c is Z.G/ . (d) In general, the map c is not surjective. The image of c is called the group of inner automorphisms and denoted Int .G/ . Conclude that Int .G/ Š G=Z.G/ . 2.7.7. Let D 4 denote the group of symmetries of the square, and N the subgroup of rotations. Observe that N is normal and check that D 4 =N is isomorphic to the cyclic group of order 2. 2.7.8. Find out whether every automorphism of S 3 is inner. Note that any automorphism ' must permute the set of elements of order 2, and an auto-
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: morphism ' is completely determined by what it does to order 2 elements, since all elements are products of 2–cycles. Hence, there can be at most as many automorphisms of S 3 as there are permutations of the three–element set of 2–cycles, namely 6; that is, j Aut .S 3 / j ² 6 . According to Exercises 2.5.13 and 2.7.6 , how large is Int .S 3 / ? What do you conclude? 2.7.9. Let G be a group and let C be the subgroup generated by all ele-ments of the form xyx ± 1 y ± 1 with x;y 2 G . C is called the commutator subgroup of G . Show that C is a normal subgroup and that G=C is abelian. Show that if H is a normal subgroup of G such that G=H is abelian, then H ³ C . 2.7.10. Show that any quotient of an abelian group is abelian. 2.7.11. Prove that if G=Z.G/ is cyclic, then G is abelian. 2.7.12. Suppose G=Z.G/ is abelian. Must G be abelian?...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern