hw17 - Let G be a group and for a ∈ G let c a : G → G...

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Modern Algebra 1 Homework 8.1 Spring 2010 Due March 24 Exercise 1. Let G be a group and suppose that | G | = p , a prime number. Prove that G = Z / h p i . Find, with proof, every subgroup of G . Exercise 2. Suppose that G is a finite group and that f : G H is a group homo- morphism. Use Lagrange’s Theorem and the First Isomorphism Theorem to prove that | G | = | ker f || Im f | . Exercise 3.
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Unformatted text preview: Let G be a group and for a ∈ G let c a : G → G denote the automorphism of G given by c a ( x ) = axa-1 . Recall that Inn( G ) = { c a | a ∈ G } and Z ( G ) = { x ∈ G | xy = yx for all y ∈ G } . Use the first isomorphism theorem to prove that G/Z ( G ) ∼ = Inn( G ). Exercise 4. Lang, II.4.28....
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This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.

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