Algebra Qualifying Examination
January, 2004
Directions:
1. Answer all questions. (Total possible is 100 points.)
2. Start each question on a new sheet of paper.
3. Write only on one side of each sheet of paper.
Policy on Misprints:
The Qualifying Exam Committee tries to proofread the exams as carefully as possible.
Nevertheless, the exam may contain a few misprints. If you are convinced a problem
has been stated incorrectly, indicate your interpretation in writing your answer. In such
cases, do not interpret the problem in such a way that it becomes trivial.
Notes:
1. All rings are unitary. All modules are unitary.
2.
is the rationals,
the reals,
the complexes, and
the integers.
Questions
1. (10 points)
Determine all groups that have exactly 3 subgroups.
2. (15 points)
Let
Z
(
G
) and Inn(
G
) denote the center of the group
G
and the group of inner
automorphisms of
G
respectively.
(i) Prove that
G/Z
(
G
) is isomorphic to Inn(
G
).
(ii) Suppose that the group Aut(
G
) of automorphisms of
G
is cyclic. Prove that
G
is abelian.
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 Spring '08
 comech
 Vector Space, Ring

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