MATH 436 Notes: Applications of Unique
Factorization.
Jonathan Pakianathan
December 13, 2005
1
Euclidean rings
We have previously discussed the basic concepts and fundamental theorems
on unique factorization in PIDs. We will provide a few examples of how
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MATH 436: Homework V.
Due in class on Wednesday, Oct 26
1. [Groups generated by 2 involutions.] Let G =< a, b > with o(a) =
o(b) = 2 and assume G is not cyclic. Let = ab.
(a) Show that aa1 = bb1 = 1 . Conclude that if K =< > then
K G. Show that aK = bK an
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MATH 436 Notes: Functions and Inverses.
Jonathan Pakianathan
September 12, 2003
1
Functions
Denition 1.1. Formally, a function f : A B is a subset f of A B with
the property that for every a A, there is a unique element b B such that
(a, b) f . The set A
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MATH 436 Notes: Subgroups and Cosets.
Jonathan Pakianathan
September 15, 2003
1
Subgroups
Denition 1.1. Given a group (G, ), a subset H is called a subgroup of G
if it itself forms a group under . Explicitly this means:
(1). The identity element e lies in
MATH 436 Notes: Rings.
Jonathan Pakianathan
November 13, 2003
1
Rings
Denition 1.1 (Rings). A ring R is a set with two binary operations +
and called addition and multiplication respectively such that:
(1) (R, +) is an Abelian group with identity element
MATH 436 Notes: Finitely generated Abelian
groups.
Jonathan Pakianathan
November 1, 2003
1
Direct Products and Direct Sums
We discuss two universal constructions in this section. We start with the
direct product construction:
Denition 1.1 (Direct Products
MATH 436 Notes: Factorization in
Commutative Rings.
Jonathan Pakianathan
December 2, 2003
1
Prime and Maximal Ideals
Throughout this section, ideal always means two-sided ideal.
The following proposition is basic and its proof is left to the reader:
Propo
MATH 436 Notes: Sylow Theory.
Jonathan Pakianathan
October 7, 2003
1
Sylow Theory
We are now ready to apply the theory of group actions we studied in the last
section to study the general structure of nite groups. A key role is played
by the p-subgroups o
MATH 436 Notes: Group Actions.
Jonathan Pakianathan
September 30, 2003
1
Group Actions
Denition 1.1. We say that a group G acts on a set X (on the left) if there
is an action G X X such that:
[A1:] e x = x for all x X.
[A2:] (g1 g2 ) x = g1 (g2 x) for all
MATH 436 Notes: Homomorphisms.
Jonathan Pakianathan
September 23, 2003
1
Homomorphisms
Denition 1.1. Given monoids M1 and M2 , we say that f : M1 M2 is a
homomorphism if
(A) f (ab) = f (a)f (b) for all a, b M1
(B) f (e1 ) = e2 where ei is the identity ele
MATH 436 Notes: Series.
Jonathan Pakianathan
October 16, 2003
1
Series
We now generalize the process of cutting up a group into smaller parts with
the concept of a series:
Denition 1.1 (Series). Fix a group G and subgroups H K G.
An ascending series conne
MATH 436 Notes: Examples of Rings.
Jonathan Pakianathan
November 20, 2003
1
Formal power series and polynomials
Let R be a ring. We will now dene the ring of formal power series on a
variable x with coecients in R. We will denote this ring by R[x].
As an
MATH 436 Notes: Chain conditions.
Jonathan Pakianathan
December 1, 2003
1
Chain conditions
We next consider sequences of ideals in a ring R.
Denition 1.1 (Chains). A sequence of (left) (right) ideals I1 I2 I3
I4 . . . is called an ascending chain of (lef
MATH 436 Notes: Ideals.
Jonathan Pakianathan
December 1, 2003
1
Ideals and Subrings
Denition 1.1 (Subring). Let (R, +, ) be a ring and S a subset of R which
is itself a ring under + and with the same multiplicative identity 1 then we
call S a subring of R
MATH 436 Notes: Cyclic groups and Invariant
Subgroups.
Jonathan Pakianathan
September 30, 2003
1
Cyclic Groups
Now that we have enough basic tools, let us go back and study the structure of
cyclic groups. Recall, these are exactly the groups G which can b
MATH 436: Homework I.
Due in class on Friday, Sep 19
1. Fix a set S. Let P (S) denote the power set of S, i.e., P (S) = cfw_A|A S.
(a) Check that P (S) is an Abelian monoid under the operation , where
A1 A2 is the intersection of the subsets A1 and A2 . W