Transcript Properties of a Group

Abstract Algebra Part 1
Cumulative Review
Text: Contemporary Abstract Algebra by J. A. Gallian,
6th edition
This presentation by:
Jeanine “Joni” Pinkney
in partial fulfillment of requirements of Master of Arts in Mathematics Education
degree
Central Washington University
Fall 2008
Picture credit:
euler totient graph
http://www.123exp-math.com/t/01704079357/
Contents:
Chapter 2. Groups
Definition and
Examples
Elementary Properties
Chapter 3: Finite Groups;
Subgroups
Terminology and
Notation
Subgroup Tests
Examples of Subgroups
Chapter 4: Cyclic Groups
Properties of Cyclic
Groups
Classifications of
Subgroups of
Cyclic Groups
Chapter 5: Permutation Groups
Definition and Notation
Cycle Notation
Properties of Permutations
Suggested Activities
Practice with Cyclic
Notation
Online Resources
provided by
text author J.A.
Gallian
Other Online Resources
Acknowledgments
Photo credit:A5, the smallest nonabelian group
http://www.math.metu.edu.tr/~berkman/466object.html
Suggested Uses of this
Presentation:
Review for final
exam for Math 461*
Review in
preparation for Math
462*
Review for
challenge exam for
course credit for
Math 461*
Independent Study
*or similar course

math cartoons from
http://www.math.kent.edu/~sather/ugcolloq.html
Definition of a Group
A Group G is a collection of elements
together with a binary operation* which
satisfies the following properties:
Closure
Associativity
Identity
Inverses
* A binary operation is a function on G
which assigns an element of G to each
ordered pair of elements in G. For
example, multiplication and addition are
binary operations.
rubic cube permutation group
http://en.wikipedia.org/wiki/Permutation_group
Classification of Groups
Groups may be Finite or Infinite;
that is, they may contain a finite
number of elements,
or an infinite number of elements.
Also, groups may be Commutative
or
Non-Commutative,
that is, the commutative property
may or may not apply to all
elements of the group.
Commutative groups are also
called Abelian groups.
“Abelian... Isn't that a one followed by a
bunch of zeros?”
- anonymous grad student in MAT
program
symmetry 6 ceiling art
http://architecturebuildingconstruction.blogspot.com/2006_03_01_archive.html
Examples of Groups
Examples of Groups:
Infinite, Abelian:
The Integers under Addition (Z. +)
The Rational Numbers without 0 under multiplication (Q*, X)
Infinite, Non-Abelian:
The General Linear Groups (GL,n), the nonsingular nxn matrices
under matrix multiplication
Finite, Abelian:
The Integers Mod n under Modular Addition (Zn , +)
The “U groups”, U(n), defined as Integers less than n and relatively
prime to n, under modular multiplication.
Finite, Non-Abelian:
The Dihedral Groups Dn the permutations on a regular n-sided
figure under function composition.
The Permutation Groups Sn, the one to one and onto functions from
a set to itself under function composition.
euler totient graph
http://www.123exp-math.com/t/01704079357/
Properties of a Group:
Closure
“If we combine any two elements in the group under the binary
operation, the result is always another element in the group.” -- Geoff
“Not necessarily another element of the group!” -- Joni
Example:
The Integers under Addition, (Z, +)
1 and 2 are elements of Z,
1+2 = 3, also an element of Z
Non-Examples:
The Odd Integers are not closed under
Addition. For example, 3 and 5 are odd
integers, but 3+5 = 8 and 8 is not an odd integer.
The Integers lack inverses under
Multiplication, as do the Rational numbers
(because of 0.) However, if we remove 0 from
the Rational numbers, we obtain an infinite
closed group under multiplication.
"members only"
http://en.wikipedia.org/wiki/index.html?curid=12686
870
Properties of a Group: Associativity
The Associative Property, familiar from
ordinary arithmetic on real numbers,
states that (ab)c = a(bc). This may be
extended to as many elements as
necessary.
For example:
In Integers,
a+(b+c) = (a+b)+c.
Caution:
In Matrix Multiplication,
(A*B)*C=A*(B*C).
The Commutative Property, also familiar
from ordinary arithmetic on real numbers,
does not generally apply to all groups!
In function composition,
f*(g*h) = (f*g)*h.
Only Abelian groups are commutative.
This may take some “getting-used-to,” at first!
This is a property of all groups.
associative loop
http://en.wikipedia.org/wiki/List_of_algebraic_structures
Properties of a Group: Identity
The Identity Property, familiar from
ordinary arithmetic on real numbers,
states that, for all elements a in G,
a+e = e+a = a.
For example,
in Integers, a+0 = 0+a = a.
In (Q*, X), a*1 = 1*a = a.
In Matrix Multiplication, A*I = I*A = A.
This is a property of all groups.
|1 0| = I
|0 1|
The Identity is Unique!
There is only one identity
element in any group.
This property is used in
proofs.
Properties of a Group:
Inverses
The inverse of an element, combined with that element, gives the
identity.
Inverses are unique. That is, each element has exactly
one inverse, and no two distinct elements have the same inverse.
The uniqueness of inverses is used in proofs.
For example...
In (Z,+), the inverse of x is -x.
In (Q*, X), the inverse of x is 1/x.
In (Zn, +), the inverse of x is n-x.
In abstract algebra, the inverse of an element a is usually written a-1.
This is why (GL,n) and (SL, n) do not include singular matrices; only
nonsingular matrices have inverses.
In Zn, the modular integers, the group operation is understood to be
addition, because if n is not prime, multiplicative inverses do not
exist, or are not unique.
The U(n) groups are finite groups under modular multiplication.
Abelian Groups
Abelian Groups are groups which have the
Commutative property, a*b=b*a for all a and b in G.
This is so familiar from ordinary arithmetic on Real
numbers, that students who are new to Abstract
Algebra must be careful not to assume that it applies
to the group on hand.
Abelian groups are named after Neils Abel, a
Norwegian mathematician.
Neils Abel postage stamp http://en.wikipedia.org/wiki/Neils_Abel
Abelian groups may be recognized
by a diagonal symmetry in their
Cayley table (a table showing the
group elements and the results of
their composition under the group
binary operation.)
This symmetry may be used in
constructing a Cayley table, if we
know that the group is Abelian.
Cayley tables for Z4 and U8
http://www.math.sunysb.edu/~joa/MAT313/hw-VIII---313.html
Examples of Abelian Groups
Some examples of Abelian groups are:

The Integers under Addition, (Z,+)

The Non-Zero Rational Numbers under
Multiplication, (Q*, X)

The Modular Integers under modular addition,
(Zn, +)

The U-groups, under modular multiplication,
U(n) = {the set of integers less than or equal to
n, and relatively prime to n}

All groups of order 4 are Abelian. There are
only two such groups: Z4 and U(4).
http://www.math.csusb.edu/faculty/susan/modular/modular.html
Non-Abelian Groups
Some examples of Non-Abelian
groups are:
Dn, the transformations on a regular nsided figure under function composition
(GL,n), the non-singular square matrices
of order n under matrix multiplication
(SL,n), the square matrices of order n
with determinant = 1under matrix
multiplication
Sn, the permutation groups of degree n
under function composition
An, the even permutation groups of
degree n under function composition
permutation group A4
http://faculty.smcm.edu/sgoldstine/origami/displaytext.html
permutation group s5
http://www.valdostamuseum.org/hamsmith/PDS3.html
D3 knot
http://www.math.utk.edu/
~morwen/3d_pics/more_
d3.html
reflections of a triangle
http://www.answers.com/topic/di
hedral-group
subgroup lattice for s3
http://www.mathhelpfor
um.com/mathhelp/advancedalgebra/22850-normalsubgroup.html