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 elements 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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 Fall '08
 EVERAGE
 Algebra, Transformations, Normal subgroup, Abelian group, 2 g, automorphism, fractional linear transformations

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