Math 6121
Fall 2013
The Cayley-Hamilton theorem for modules
Let R be a commutative ring with identity. We begin by recalling the
following basic property of determinants:
Lemma 1. Let A be an n n matrix with entries in R and let A be the
adjoint matrix of
Math 6121
Fall 2013
A proof of Sylows theorems
In this handout, we give proofs of the three Sylow theorems which are
slightly dierent from the ones in the book. Recall the following lemma from
the handout on Cauchys theorem.
Lemma. Let p be a prime number
1. P OLYA ENUMERATION P lya enumeration is useful in counting problems where symmetry exists. o Question 1.1. Fix n. What is the number of essentially different necklaces that can be made with n beads of two different colors? Here, we say that two neckla
Math 6121
Fall 2013
A proof of Cauchys theorem
We give McKays clever proof of Cauchys theorem.
Lemma. Let p be a prime number, and let G be a p-group (a nite group of
order pk for some k 1) acting on a nite set S . Let Fix be the set of xed
points of the
Math 6121
Fall 2013
HW assignment #5 (Due Thursday, September 26th)
1. Write down a complete list of all abelian groups of order 270 (up to
isomorphism) using (a) invariant factors and (b) elementary divisors. Be
sure to specify which groups correspond to
Math 6121
Fall 2013
HW assignment #4 (Due Thursday, September 19)
1. Determine all nite groups which have exactly two conjugacy classes.
2. a. Let G be a nite group, and let H be a proper subgroup of G. Prove
that the union of all conjugates of H is a pro
Math 6121
Fall 2012
HW assignment #1 (Due Thursday, August 29)
1. (1.1 #22) If x, y are conjugate elements of a group G (this means that
y = g 1 xg for some g G), prove that x and y have the same order.
Deduce that |ab| = |ba| for all a, b G.
2. (1.2 # 4,
Math 6121
Fall 2013
HW assignment #3 (Due Thursday, September 12)
1. Let G be a group, and let H be a subgroup of G.
a. Verify that G acts by left multiplication on the left coset space G/H .
b. Show that this action is transitive, and compute the stabili
Math 6121
Fall 2013
Finitely Generated Abelian Groups
We give a brief account of the structure theory of nitely generated
abelian groups.
Recall that a free abelian group is a group G which is isomorphic to
Zn for some positive integer n, called the rank