# hw6 - Math 113 Homework 6 selected solutions Fraleigh 14.34...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

hw6 - Math 113 Homework 6 selected solutions Fraleigh 14.34...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online