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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 ) = axa1 . 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.
 Spring '10
 RyanDaileda
 Algebra

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