Homework #10. Due Thursday, November 18th
Reading:
1. For this assignment: Sections 4.4, 4.5 and the second part of 4.1 (even/odd
permutations and conjugacy classes in Sn ) + online notes
2. For the c
Direct products and sums of groups.
1. Direct products. Let A and B be two groups. The direct product of A
and B , denoted by A B , is dened as follows:
As a set, A B is simply the direct product of t
23. Quotient groups II 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time. Theorem (FTH). Let G, H be groups and : G H a hom
21. Permutation groups II 21.1. Conjugacy classes. Let G be a group, and consider the following relation on G: given f, h G, we put f h there exists g G s.t. h = gf g -1 . Thus, in the terminolgy from
24. Rings 24.1. Definitions and basic examples. Definition. A ring R is a set with two binary operations + (addition) and (multiplication) satisfying the following axioms: (A0) (A1) (A2) (A3) (A4) (M0
25. Ideals and quotient rings We continue our study of rings by making analogies with groups. The next concept we introduce is that of an ideal of a ring. Ideals are ring-theoretic counterparts of nor
26. Examples of quotient rings In this lecture we will consider some interesting examples of quotient rings. First we will recall the definition of a quotient ring and also define homomorphisms and is
27. Fields from quotient rings In Lecture 26 we have shown that the quotient ring R[x]/(x2 + 1)R[x] is isomorphic to C, so, in particular, it is a field, while R[x]/(x2 - 1)R[x] is not a field. The re
15. Isomorphisms (continued)
We start by recalling the notions of an isomorphism between groups and
the notion of isomorphic groups.
Definition. Let G and G0 be groups.
(a) A map : G G0 is called an i
18. Lagrange Theorem and Classification of groups of small
order
18.1. Lagrange Theorem and its immediate consequences.
Lagrange Theorem. Let G be a finite group and H a subgroup of G. Then
|H| divide
16. Homomorphisms
16.1. Basic properties and some examples.
Definition. Let G and H be groups. A map : G H is called a
homomorphism if
(xy) = (x)(y) for all x, y G.
Example 1. Let G = (Z, +) and H = (
16A. Direct products and Classification of Finite Abelian
Groups
16A.1. Direct products.
Definition. Let G and H be groups. Their direct product is the group
G H defined as follows. As a set G H = cfw
22. Quotient groups I 22.1. Definition of quotient groups. Let G be a group and H a subgroup of G. Denote by G/H the set of distinct (left) cosets with respect to H. In other words, we list all the co
Homework #12. Due Tuesday, December 7th
Reading:
For this assignment: Sections 5.1, 6.1, 6.2.
Problems:
Problem 1: Let R be a commutative ring with 1 and I an ideal of R.
(a) Suppose that 1 I . Prove
Homework #11. Due Thursday, December 2nd
Reading:
1. For this assignment: Sections 4.6 and 5.1 + online notes from Lectures 22
and 23.
2. For the class on Nov 23: Section 5.1 (rings)
3. For the class
A little memo on injective, surjective and bijective functions
1. Formally, a function is dened as follows (see [GG, 1.2]). Given two sets
A and B , a function from A to B is a subset f of the Cartesi
A. Even and odd permutations (brief summary)
Recall that a transposition is a cycle of length 2.
Lemma A.1. Any permutation f Sn can be written as a product of
transpositions.
Proof. Since any permuta
Homework #2. Due Thursday, September 9th, in class
Reading:
1. For this assignment: Sections 2.3 and 2.4 (up to Denition 2.13).
2. Before the class on Tuesday, Sep 7th: the rest of Section 2.4. Before
Homework #3. Due Thursday, September 16th, in class
Reading:
1. For this assignment: Sections 2.4 and 2.5, up to Theorem 2.26.
2. Before the class on Tuesday, Sep 14th: the rest of Section 2.5, Sectio
Homework #1. Due TUESDAY, August 31st, in class
Reading:
1. For this homework assignment: Sections 2.1, 2.2 and 1.2. A good supplement to Section 1.2 is provided by Wikipedia - look at the articles In
Homework #5. Due Thursday, September 30th, in class
Reading:
For this homework assignment: Sections 2.6, 3.1 and 3.2.
Plan for next weeks classes: 3.2 (nish), 3.3, 3.4 (start)
To hand in:
Problem 1: S
Homework #4. Due Thursday, September 23rd, in class
Reading:
For this homework assignment: Sections 1.7, 2.6 and the end of 2.5.
For next weeks classes: Sections 3.1 and 3.2
To hand in:
Problem 1: Let
Homework #6. Due Thursday, October 14th
Reading:
For this homework assignment: Sections 3.3 and 3.4 (up to page 167)
Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168-171). Also
review the
Homework #8. Due Thursday, October 28th
Reading:
1. For this assignment: Section 3.6 + online supplement on direct products.
2. for Tuesdays class: Section 4.1. Read at least up to Example 7 on page
1
Homework #7. Due Thursday, October 21st
Reading:
1. For this homework assignment: Sections 3.4 and 3.5.
2. For next weeks classes: Section 3.6.
Problems:
Problem 1: Recall Theorem 14.1 stated in class
Homework #9. Due Thursday, November 4th
Reading:
1. For this homework assignment: Section 4.1 and the part of 4.4 dealing
with the statement and applications of Lagrange theorem.
2. For Tuesdays class
17. Symmetric groups
Fix an integer n > 1, and let Sn be the set of all bijective functions
f : cfw_1, . . . , n cfw_1, . . . , n. As discussed in Lecture 10, Sn is a group
with respect to composition