MATH 436 Notes: Applications of Unique
December 13, 2005
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.
September 12, 2003
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.
September 15, 2003
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.
November 13, 2003
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
November 1, 2003
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
December 2, 2003
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:
MATH 436 Notes: Sylow Theory.
October 7, 2003
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.
September 30, 2003
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.
September 23, 2003
Denition 1.1. Given monoids M1 and M2 , we say that f : M1 M2 is a
(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.
October 16, 2003
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.
November 20, 2003
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].
MATH 436 Notes: Chain conditions.
December 1, 2003
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.
December 1, 2003
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
September 30, 2003
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