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Modern Algebra 1
Homework 8.3
Spring 2010
Due March 24
Exercise 10.
Let
f
:
R
×
R
→
R
be given by
f
(
x,y
) = 2
x

3
y
.
a.
Prove that
f
is a surjective homomorphism.
b.
Find ker
f
and describe it and its cosets geometrically.
c.
The First Isomorphism Theorem implies that
f
induces an isomorphism
f
: (
R
×
R
)
/
ker
f
→
R
. Describe this isomorphism geometrically.
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Unformatted text preview: Exercise 11. Let G be a ﬁnite group, let H ≤ G and let K C G . Prove that if  H  is relatively prime to [ G : K ] then H ≤ K . [ Hint: Given a ∈ H , by considering the orders of a in H and aK in G/K , show that h a i ≤ K .] Exercise 12. Lang, II.4.30....
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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, Sets

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