Syllabus for Preliminary Exam for 631-632 Algebra I, II
Linear algebra: vector spaces, linear transformations, eigenvectors and
diagonalization, Jordan canonical form, bilinear forms and inner product spaces,
normal operators.
Groups: cosets, quotient gro
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Algebra Ph.D. Preliminary Exam
January 12, 2010
1. Recall that a subgroup H of a group G is called characteristic if (H ) H for every automorphism
of G.
(a) Prove that characteristic subgroups are always normal.
(b) Let P be a p-Sylow subgroup of a
Ph. D. Algebra Preliminary Exam August 23, 2010
Show the work you do to obtain an answer. Give reasons for your answers.
There are 10 questions. Do all parts of all questions. Each question is worth 10
points. Q is the eld of rational numbers. C is the el
ALGEBRA PHD PRELIMINARY EXAM, 9 JANUARY 2009
1. (10 points) Let G be a group of order 132 = 22 3 11. Prove that G is not simple.
2. (10 points) Let H and K be normal subgroups of a group G , and assume that G = HK .
Prove that there is an isomorphism
G /(
Ph. D. Algebra Preliminary Exam Tuesday, August 25, 2009
Show the work you do to obtain an answer. Give reasons for your answers.
There are 10 questions. Do all parts of all questions. Each question is worth
10 points. When a question has two parts (a) an
Algebra Preliminary Examination, January 10, 2008
Print name:
Show your work, give reasons for your answers, provide all necessary proofs and
counterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100
points. Check that y
Algebra Preliminary Examination, August 19, 2008
Print name:
Show your work, give reasons for your answers, provide all necessary proofs
and counterexamples. There are 10 problems on 15 pages. Each problem is worth 10
points for a total of 100 points. Ple
Ph. D. Algebra Preliminary Exam Wednesday, January 10, 2007
Show the work you do to obtain an answer. Give reasons for your answers.
There are 10 questions. Do all parts of all questions.
Notations: C (x) is the centralizer of a group element x, , is the
Ph. D. Algebra Preliminary Exam Monday, August 20, 2007
Show the work you do to obtain an answer. Give reasons for your answers.
There are 10 questions. You should answer each question.
1. (10 points) Let G be a group of order p2 for some prime number p.