College Algebra Exam Review 136

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

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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-
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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?...
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