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Fall 2009
MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #2
Week
#1: September 28th  October 4th.
Topics
: Universal property of quotients, the isomorphism theorems, subgroups of
quotients, direct products, structure of ﬁnitely generated abelian groups, the dual
group of characters, orthogonality relations, group actions, orbits, stabilizers, the
class equation, Cauchy’s theorem on existence of elements of prime order.
Read
: GT [1926] and GT [5057]
Problems
due Friday, October 2nd, at 4:00 pm in Martin Luu’s mailbox:
•
Problem 1
:
Show that
S
n
is generated by the tranpositions
s
i
= (
i,i
+ 1)
,
and that they satisfy
1
the braid group relations: That is,
s
i
s
j
=
s
j
s
i
when

i

j

>
1
, and
s
i
s
j
s
i
=
s
j
s
i
s
j
when

i

j

= 1
.
•
Problem 2
:
Is
D
4
isomorphic to
Q
8
?
•
Problem 3
:
Show that any subgroup of
Z
is of the form
n
Z
, for a unique
nonnegative integer
n
. Given positive integers
m
and
n
, what are the
positive generators of the two subgroups
m
Z
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This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.
 '09
 CLAUSSORENSEN
 Algebra

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