8/11/2019 MATH 113 Textbook Notes

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1.1 

Basic Axioms and Examples

  Basics

o  Binary operation on a set

o  Associativity and Commutativity of a binary operation on a set

o 

A set is closed under an operation if….. 

 

Definition of a Groupo  A group is an ordered pair (G,*) G a set and * a binary operation that

  Is associative, has an identity element and has inverses

o  A group is abelian if it is commutative as well

  The orderof an element in a group is defined as ……… 

o 

We define something as having infinite order ………… 

Interesting Exercises: 5, 8-9, 30, 32

1.2 

Dihedral Groups

 

D2n is set of symmetries of a regular n-gon, called the dihedral group of order 2n

  “r” for clockwise rotation, and “s” for reflection about line from vertex 1 to the origin 

  rn = 1, s2 = 1, ris = sr-i 

  Generators and relations

Interesting Exercises: All boring

1.3 

Symmetric Groups

  Set of all bijections of a set to itself (1,2,3…n) with binary operation of composition 

o  Denoted as Sn, the symmetric group of degree n

o  Sn has an order of n!

 

Cycle decomposition algorithmo  Length of a cycle, m-cycle, disjoint cycles

o  Cycles of length 1 are omitted

o  Cycles can be shifted

  Sn is a non-abelian group of n \geq 3

  Disjoint cycles commute

  The order of a permutation is the l.c.m of the lengths of the cycles in the composition

Interesting Exercises: didn’t do 11-20

1.6 

Homomorphisms and Isomorphisms

 

Definition of Homomorphism  Definition of Isomorphism

o 

Note: A map is a bijection IFF its invertible

o  Isomorphisms preserve elements of group structure (e.g. commutativity)

o  Classification Theorem: Any non-abelian group of order 6 is isomorphic to S3 

o  Up to isomorphism there are two groups of order 6, S3 and Z/6Z

  If phi: G -> H is an isomorphism:

o  |G| = |H|

8/11/2019 MATH 113 Textbook Notes

http://slidepdf.com/reader/full/math-113-textbook-notes 2/2

o  G is abelian iff H is abelian

o  For all x $\in$ G, |x| = |phi(x)|

Interesting Exercises: 4-6, 13-14, 19-20, 23-24

1.7 

Group Actions (NEEDS WORK)

  Definition of Group Action

  “Two Important Facts” and Proof  

  Definition of a “Faithful” Group Action (injectivity)

  Definition of the kernel of a group action

Interesting Exercises: 4 (skip most of these, i dont like them)

2.1 Subgroups: Definition and Examples

 

The Subgroup Criterion

o 

Don’t forget the nonempty part! 

Interesting Exercises: 4,5,6,12-15

2.2Centralizers and Normalizers, Stabilizers and Kernels

  Let A be a subset of G

  The Centralizer of A in G is g \in G s.t ga = ag \forall a \in A or CG(A)

  The Center of Gis the set all G that commutes with all G \in G or CG(G) 

  The Normalizer of A in G is g \in G s.t. gA-1g = A \forall a \in A or NG(A) 

 

The Stabilizer of s \in G is the set of all g \in G s.t gs = s   The Kernel of G is the set of all g \in G s.t.gs = s \forall s \in S 

Interesting Exercises: 6-11

2.3 Cyclic Groups 

  A group is cyclic if it can be generated by some single element

  Cyclic groups of the same order are isomorphic

Interesting Exercises: 12,13,15,16

Define all the words on the list and give an example.

Skim Prelims, 1.1,1.2,1.3,1.6, 2.1 and the exercises

Go over 1.7, 2.2-2.4 carefully!

Notes (Braids!)s