MATH 516
Midterm
Radford
WEDNESDAY 10/24/01
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your
exam booklet. (3) Show your work. (4) There are ve questions on this exam. (5) Each
problem counts 20 points. (6) You are expect
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
SPRING 2009
SOLUTIONS
Algebra
A1. Show that if H is subgroup of a group G having nite index n then there is a normal
subgroup K of G which is a subgroup of H and such that |G : K| n!. (Hint: consi
PURE MATH OPTION Solutions, Spring 2012
Algebra
A1. Let G be a simple group of order 60.
(a) Prove that G contains exactly six subgroups of order 5.
(b) Prove that G is isomorphic to a subgroup of A6 , the alternating group on six letters.
(This can be us
PURE MATH OPTION, Fall 2012
Algebra
A1. Show that every group of order 294 = 49 2 3 has a normal subgroup of order 147.
Solution. Let G be a group of order 294 and n7 be the number of Sylow 7-subgroups of
G. By Sylow theorems n7 1 mod 7 and n7 |6. Thus n7
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
FALL 2009
SOLUTIONS
Algebra
A1. Assume H and K are subgroups of a group G. Show that HK is a subgroup of G if
and only if HK = KH.
Solution: Assume rst that HK = KH and let a, b HK. We prove ab1 H
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
FALL 2008
Full points can be obtained for correct answers to 8 questions. Each numbered question
(which may have several parts) is worth the same number of points. All answers will be
graded, but
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
FALL 2010
Full points can be obtained for correct answers to 8 questions. Each numbered question
(which may have several parts) is worth 20 points. All answers will be graded, but the score
for th
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
FALL 2007
Full points can be obtained for correct answers to 8 questions. Each numbered question
(which may have several parts) is worth the same number of points. All answers will be
graded, but
MASTERS EXAMINATION IN MATHEMATICS
PURE MATH OPTION, Spring 2013
Full points can be obtained for correct answers to 8 questions. Each numbered question
(which may have several parts) is worth 20 points. All answers will be graded, but the score
for the ex
MASTERS EXAMINATION IN MATHEMATICS
PURE MATHEMATICS OPTION
SPRING 2010
SOLUTIONS
Algebra
A1. Let F be a nite eld. Prove that F [x] contains innitely many prime ideals.
Solution: The ring F [x] of polynomials with coecients in a eld F is a P.I.D.
Each prim
Math 516
Fall 2008
Radford
Written Homework # 5 Solution
12/03/08
1. (20 points) Note that if R1 , . . . , Rn are nite Boolean rings then the direct product
R1 Rn is a Boolean ring also. Thus part (c) characterizes nite Boolean rings.
(a) (8 pts) Let a, b
Math 516
Fall 2006
Solution to the Final Examination
Radford
12/17/06
Name (PRINT)
(1) Return this exam copy. (2) Write your solutions in your exam booklet. (3) Show your
work. (4) There are eight questions on this exam. (5) Each question counts 25 points
MATH 516
Hour Exam
Radford
10/17/08
Name (print)
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your
exam booklet. (3) Show your work. (4) There are four questions on this exam. (5) Each
question counts 25 points. (6) You ar
MATH 516
Hour Exam
Radford
10/13/06 10AM
Name (print)
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your
exam booklet. (3) Show your work. (4) There are ve questions on this exam. (5) Each
question counts 20 points. (6) You
MATH 516
Hour Exam Solution
Radford
10/23/08
Name (print)
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your
exam booklet. (3) Show your work. (4) There are four questions on this exam. (5) Each
question counts 25 points. (
Basic Examples of Rings
10/28/08 Radford
Let S be be a non-empty set and let G be a semigroup. We dene a
binary operation on the set Fun (S, G) of all functions f : S G by
pointwise multiplication, that is
(f g )(s) = f (s)g (s)
(1)
for all f, g Fun (S, G
Math 516
Fall 2008
Radford
Written Homework # 1 Solution
10/13/08
1. (20 points)
(a) (8 pts) This is straightforward. am+0 = am = am e = am a0 ; thus the formula holds when n = 0.
Suppose n 0 and the formula holds. Then am+(n+1) = a(m+n)+1 = am+n a = (am
Math 516
Fall 2008
Radford
Written Homework # 2 Solution
10/09/08
1. (20 points) The challenge of part (a) is not to fall asleep. A part of basic algebra is checking
mundane details. Let f, g, h G .
(a) (9 pts) Let i I . Since the binary operation in Gi i
Math 516
Fall 2008
Radford
Written Homework # 3 Solution
12/01/08
Here is the basis for a solution to the rst two problems.
Lemma 1 Suppose G is a group, p is a positive prime, and G has s cyclic subgroups of order p.
Then the number of elements of G of o
Math 516
Fall 2008
Radford
Written Homework # 4 Solution
12/02/08
1. (20 points) For f G = Fun(S, G) set Sf = cfw_s S | f (s) = 0. This set is
sometimes called the support of f . Let g G also. Observe that
Sf = Sf
(1)
Sf +g Sf Sg .
(2)
and
To see that lat