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PMath 336: Introduction to Group Theory
Midterm 2
June 24, 2008
Instructions
Solve all ﬁve questions. Write your answers in the notebook provided. Include full arguments in the answer.
Don’t forget to write your name and id number on the notebook. Simple calculators are allowed, but
discouraged. No additional material is allowed. The test duration is 50 minutes.
Good luck!
Notation
D
n
:
The group of symmetries of the regular
n
gon
S
n
:
The group of permutations of 1
,...,n
R
*
:
The group of nonzero real numbers under multiplication
Z
n
:
The group
{
0
,...,n

1
}
with addition modulo
n
U
n
:
The group of positive integers less than
n
and prime to
n
, with multiplication mod
n
.
Q
(quaternions):
The subgroup of
GL
2
(
C
) generated by
i
=
±
0
ı
ı
0
²
and
j
=
±
0 1

1 0
²
.
The Exam
1. Let
G
and
H
be two groups.
(a)
[10]
Write down the deﬁnitions of the following notions:
Homomorphism
from
G
to
H
,
kernel
of a
homomorphism,
image
of a homomorphism
(b)
[10]
Prove that the kernel and the image of a homomorphism are subgroups
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This note was uploaded on 04/27/2010 for the course PMATH 336 taught by Professor Moshekamensky during the Spring '08 term at Waterloo.
 Spring '08
 MOSHEKAMENSKY

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