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Unformatted text preview: Math 113 Homework # 6, selected solutions Fraleigh 14.34. To prove that H is normal, let a ∈ G ; we will show that aHa- 1 = H . We know that aHa- 1 is a subgroup of G with the same cardinality as H , since aHa- 1 is the image of H under the isomorphism i a : G → G . Since we assumed that G has only one subgroup of this cardinality, it follows that aHa- 1 = H . Fraleigh 14.37. This problem adds a layer of abstraction to what we have usually been doing, because now automorphisms of a group are being made into a new, more abstract group. A similar process of abstraction happens frequently in mathematics. But once you understand what the problem means, it is not too hard to solve. (a) Let Aut( G ) denote the set of automorphisms of G . We need to show that this is a group under composition. We know that this is closed under composition because from previous homework problems, the composition of two bijections is a bijection and the composition of two homomorphisms is a homomorphism. We know that composition of functions is associative. The identity map id G : G → G is an auto- morphism which serves as the identity element in the group Aut( G ), since composing any function with the identity map gives the same function. Finally every element has an inverse, because if f ∈ Aut( G...
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- Fall '08
- Math, Ri, commutator subgroup