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 the sets A and B , that is,
A B is the set of ordered pa
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 cosets of the form gH (with g G) without repetitions and
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 homomorphism. Then G/Ker (G). ( ) = Proof. Let K = Ker and
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 Lecture 20, f h h is a conjugate of f . Definition. Th
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) (M1) (D1) (D2) R is closed under addition addition is
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 normal subgroups. Recall that one of the main reasons why
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 isomorphisms of rings. Definition. Let R be a commutative
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 reason we did not get a field in the second case is clear
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 isomorphism if
(i) is bijective
(ii) preserves group ope
18. Lagrange Theorem and Classification of groups of small
18.1. Lagrange Theorem and its immediate consequences.
Lagrange Theorem. Let G be a finite group and H a subgroup of G. Then
|H| divides |G|.
We will prove Lagrange Theorem next week. In thi
16.1. Basic properties and some examples.
Definition. Let G and H be groups. A map : G H is called a
(xy) = (x)(y) for all x, y G.
Example 1. Let G = (Z, +) and H = (Zn , +) for some n > 1. Define
: G H by (x) = [x]. The
16A. Direct products and Classification of Finite Abelian
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_(g, h) : g G, h H is just
the usual Cartesian product
Homework #12. Due Tuesday, December 7th
For this assignment: Sections 5.1, 6.1, 6.2.
Problem 1: Let R be a commutative ring with 1 and I an ideal of R.
(a) Suppose that 1 I . Prove that I = R.
(b) Suppose that R is a eld. Prove that I =
Homework #11. Due Thursday, December 2nd
1. For this assignment: Sections 4.6 and 5.1 + online notes from Lectures 22
2. For the class on Nov 23: Section 5.1 (rings)
3. For the class on Nov 30: Section 6.1 (ideals)
4. For the class on Dec
Homework #10. Due Thursday, November 18th
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 classes on Nov 16 and 18: Section 4.6 (quotient groups)
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 Cartesian product
A B with the following property: for every a
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
Proof. Since any permutation can be written as a product of disjoint cycles, it
Homework #2. Due Thursday, September 9th, in class
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 the
class on Thursday, Sep 9th: Section 2.5.
Homework #3. Due Thursday, September 16th, in class
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, Section 2.6.
Before the class on Thursday, Sep 16th: Section
Homework #1. Due TUESDAY, August 31st, in class
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 Injective
function, Surjective function and Bijective fun
Homework #5. Due Thursday, September 30th, in class
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: Section 3.1: 2, 4 (page 142). In both problems, if G is
Homework #4. Due Thursday, September 23rd, in class
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 A be a set and an equivalence relation on A. Recall th
Homework #6. Due Thursday, October 14th
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 notion of a bijective mapping (1.2).
To hand in:
Homework #8. Due Thursday, October 28th
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
3. for Thursdays class: Read the part o
Homework #7. Due Thursday, October 21st
1. For this homework assignment: Sections 3.4 and 3.5.
2. For next weeks classes: Section 3.6.
Problem 1: Recall Theorem 14.1 stated in class:
Theorem 14.1. Let G be a nite cyclic group, n = |G| a
Homework #9. Due Thursday, November 4th
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: Section 4.4 (cosets)
3. For Thursdays class: Section
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. The groups Sn are called symmetric groups,