AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 11
October 21, 2009
1
Homomorphisms
Reading: Gallian Ch. 10
Denition: like isomorphism, but without bijection requirement
Examples:
Domain
Z
Zn
Rn
ZZ
Sn
R
Z3 Z5
G
G
Range
Zn
Zd
Rn
Z
cfw_1
C
Z
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 6
September 30, 2009
1
Permutation Groups: Basics
Reading: Gallian Chapter 5
A permutation group on a set A is a subgroup of Sym (A) (the set of permutations of A under
composition).
Examples
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 7
October 57, 2009
Reading: Gallian Chapter 6
1
Isomorphisms
Q: When are two groups the same up to the names of elements?
Z2 and an arbitrary group on two elements.
Innite cyclic group and Z.
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 5
September 28, 2009
1
Cyclic Groups & Cryptographic Applications
Reading: Gallian Chapter 4.
Classication of Subgroups of Cyclic Groups (Thms 4.2, 4.3) and Corollaries.
Example: subgroups of
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 4
September 23, 2009
1
Subgroups & Cyclic Groups
Gallian Chapters 34.
Denition of a subgroup.
cfw_0 cfw_even integers Z Q R C under addition.
Dn S n .
Subgroup tests (Gallian Thms 3.1, 3.2,
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 2
September 16, 2009
1
Primes and Factorization
Reading: Gallian Chapter 0.
In contrast to Gallian, we allow negative numbers to be prime. We dene p Z \ cfw_0, 1 to
be prime if the only diviso
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 3
September 21, 2009
1
More Groups
Reading: Gallian Chapters 1 & 2.
Group of Units modulo n (Gallian Example 2.11)
cfw_a Zn : gcd(a, n) = 1 under multiplication modulo n.
Gallian notation: U
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 8
October 14, 2009
1
Automorphisms
Reading: Gallian Chapter 6
Denition
Example: Aut(Zn ).
Def: x, y G are conjugates if y = axa1 for some a G. An equivalence relation.
Inner automorphisms:
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 9
October 19, 2009
1
Orbits and Stabilizers
Reading: Gallian Chapter 7
Defs of stabG (s), orbG (s) for G Sym (S ) and s S .
Orbit-Stabilizer Theorem (Thm. 7.3)
Proof:
For , G, (s) = (s) i st
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 14
November 4, 2009
1
Ideals
Reading: Gallian Ch. 14
Goal: ring-theoretic analogue of normal subgroup, something we can mod out (set to zero)
to get a factor ring.
Normal subgroups: since aa1
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 15
November 9, 2009
Reading: Gallian Ch. 15.
1
Homomorphisms
Denitions of ring homomorphism and ring isomorphism.
Examples and non-examples:
: Z Zn , (x) = x mod m.
: Zmn Zm Zn , (x) = (x m
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 13
November 2, 2009
1
Rings
Reading: Gallian Chs. 12 & 13
Study sets with two related operations (addition and multiplication).
Def: R with two binary operations +, is a ring if it has the fo
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 12
October 28, 2009
1
Computational Group Theory
Computational Questions about Groups:
Given a group G, compute |G|.
Given a group G, is G abelian? cyclic?
Given a group G and a subgroup H G
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 10
October 26, 2009
Reading: Gallian Chapter 9
1
Normal Subgroups
Motivation:
Cosets of nZ in Z (a + nZ) equivalence classes modulo n ([a]n )
These form a group under addition, isomorphic to
AM 106/206: Applied Algebra
Prof. Salil Vadhan
Lecture Notes 1
September 2,9,14, 2009
1
The Integers
Reading: Gallian Chapter 0.
Three forms of induction: Well-ordering Principle, Standard Induction (Thm 0.4), Strong
Induction (Thm 0.5).
Induction usua