Name:
Exam I
Instructions. Answer each of the questions on your own paper and put your name on
each page of your paper.
1. Give an example of each of the following. No proofs are required for this exercise only.
(a) A nonabelian group of order 12.
Solutio
Exercise Set 9
Math 7200
Due: December 1, 2006
Do the following exercises from the text:
Pages
179181
337340
Exercises
50, 52, 58, 59
3, 13 (a), (b), (c); 22, 30, 31
These exercises are primarily related to the computation and use of the Smith normal form
Name:
Final Exam
1. Let R be an integral domain and let M be an R-module. Give the denition of each
of the following terms:
(a) M is a free R-module.
Solution. The R-module M if free if it has a basis. (Page 129)
(b) M is a cyclic R-module.
Solution. M is
p45 #30 1. Suppose that H is a normal in G and suppose that [G : H] = n. We see that the quotient group G/H has order n. In particular we must have an = e for any a G/H (i.e. an H). 2. Take a subgroup generated by a transposition in S3 . p45 #31 Done in c
Name:
Exam II
Instructions. Answer each of the questions on your own paper and put your name on each
page of your paper.
The following denitions are provided for your convenience: If R is an integral domain, recall
that an element a of R is a unit if ab =
p45 #12
Let : H G be a homomorphism and let a H with o(a) < . One can
restrict to < a >= K and call this restriction. Then it is well known that
Im( ) K/ker( )
=
(0.1)
The result follows by taking cardinals.
If now is an isomorphism from H to G, then is a
p45 #30
1. Suppose that H is a normal in G and suppose that [G : H ] = n.
We see that the quotient group G/H has order n. In particular we
must have an = e for any a G/H (i.e. an H ).
2. Take a subgroup generated by a transposition in S3 .
p45 #31
Done in
Exercise Set 4 Solutions
Math 7200
Due: September 22, 2006
From the text (pages 45 48): 49, 50, 52.
1. Let G act on a set X . Assume that y = gx where g G and x, y X . Prove that the
stabilizers G(x) and G(y ) are conjugate subgroups of G.
Solution. Since
Exercise Set 5
Math 7200
Due: October 13, 2006
From the text (pages 98 106): 9, 14, 18
1. Recall that if R is a ring, then R denotes the group of units of R. Also, if n is a
natural number, then the Euler phi function is dened as
(n) = |Z | = |cfw_m : 1 m
Exercise Set 6
Math 7200
Due: October 20, 2004
From the text (pages 98 106): 21, 22
1. If I = 1 + 2i is the principal ideal generated by 1 + 2i in the ring of Gaussian integers
Z[i], then show that Z[i]/I is a nite eld, and nd its order.
Solution. Dene a
Exercise Set 7
Math 7200
Due: October 27, 2006
1. Recall from class that if V is a vector space over a eld F and T : V V is a linear
transformation, then the vector space V is made into an F[X ] module VT by dening
the scalar multiplication
f (X )v = f (T
Exercise Set 8
Math 7200
Due: November 17, 2006
Do the following exercises from Chapter 3 of the text (Pages 174181): 8, 9, 17, 29, 43,
47
8. Show that Q is a torsion-free Z-module that is not free.
Solution. Q is torsion free as a Z module since Q is a e
Exercise Set 9
Math 7200
Due: December 1, 2006
Do the following exercises from the text:
Pages
179181
337340
Exercises
50, 52, 58, 59
3, 13 (a), (b), (c); 22, 30, 31
These exercises are primarily related to the computation and use of the Smith normal form
Exercise Set 10
Math 7200
Due: December 8, 2006
1. (a) Show that the matrix A M3 (F) (F a eld) is uniquely determined up to similarity
by the characteristic polynomial cA (X ) and the minimal polynomial mA (X ).
Solution. A matrix A M3 (F) determines a li
Name:
Final Exam
1. Let R be an integral domain and let M be an R-module. Give the denition of each
of the following terms:
(a) M is a free R-module.
Solution. The R-module M if free if it has a basis. (Page 129)
(b) M is a cyclic R-module.
Solution. M is
p45 #12 Let : H G be a homomorphism and let a H with o(a) < . One can restrict to < a >= K and call this restriction. Then it is well known that (0.1) Im() K/ker() =
The result follows by taking cardinals. If now is an isomorphism from H to G, then is an