This document provides a summary and review of key concepts from Abstract Algebra Part 1, including: - Groups and their properties of closure, associativity, identity, and inverses - Examples of finite and infinite, abelian and non-abelian groups - Subgroups, including cyclic subgroups, tests for subgroups, and examples - Additional concepts like the order of an element, conjugation, and the center of a group

1. Math 461
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/

2. Chapter 2. GroupsChapter 2. Groups
Definition andDefinition and
ExamplesExamples
Elementary PropertiesElementary Properties
Chapter 3: Finite Groups;Chapter 3: Finite Groups;
SubgroupsSubgroups
Terminology andTerminology and
NotationNotation
Subgroup TestsSubgroup Tests
Examples of SubgroupsExamples of Subgroups
Chapter 4: Cyclic GroupsChapter 4: Cyclic Groups
Properties of CyclicProperties of Cyclic
GroupsGroups
Classifications ofClassifications of
Subgroups ofSubgroups of
Cyclic GroupsCyclic Groups
Contents:
Chapter 5: Permutation GroupsChapter 5: Permutation Groups
Definition and NotationDefinition and Notation
Cycle NotationCycle Notation
Properties of PermutationsProperties of Permutations
Suggested ActivitiesSuggested Activities
Practice with CyclicPractice with Cyclic
NotationNotation
Online ResourcesOnline Resources
provided byprovided by
text author J.A.text author J.A.
GallianGallian
Other Online ResourcesOther Online Resources
AcknowledgmentsAcknowledgments
Photo credit:A5, the smallest nonabelian group
http://www.math.metu.edu.tr/~berkman/466object.html

3. 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
Suggested Uses of this
Presentation:

4. 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

5. 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://architecture-
buildingconstruction.blogspot.com/2006_03_01_archive.html

6. 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/

7. Properties of a Group:
Closure
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.
“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
"members only"
http://en.wikipedia.org/wiki/index.html?
curid=12686870

8. 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.
In Matrix Multiplication,
(A*B)*C=A*(B*C).
In function composition,
f*(g*h) = (f*g)*h.
This is a property of all groups.
Caution:
The Commutative Property, also familiar
from ordinary arithmetic on real numbers,
does not generally apply to all groups!
Only Abelian groups are commutative.
This may take some “getting-used-to,” at first!
associative loop
http://en.wikipedia.org/wiki/List_of_algebraic_structures

9. 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.
The Identity is Unique!
There is only one identity
element in any group.
This property is used in
proofs.
|1 0|
|0 1|
= I

10. 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.

11. 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.
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.
Neils Abel postage stamp http://en.wikipedia.org/wiki/Neils_Abel
Cayley tables for Z4 and U8
http://www.math.sunysb.edu/~joa/MAT313/hw-VIII---313.html

12. 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).
Examples of Abelian Groups
http://www.math.csusb.edu/faculty/susan/modular/modular.html

13. Non-Abelian Groups
Some examples of Non-Abelian
groups are:
Dn
, the transformations on a regular n-
sided 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
subgroup lattice for s3
http://www.mathhelpfor
um.com/math-
help/advanced-
algebra/22850-normal-
subgroup.html
reflections of a triangle
http://www.answers.com/topic/di
hedral-group
D3 knot
http://www.math.utk.edu/
~morwen/3d_pics/more_
d3.html
permutation group A4
http://faculty.smcm.edu/sgoldstine/origami/displaytext.html
permutation group s5
http://www.valdostamuseum.org/hamsmith/PDS3.html

14. Finite Groups and
Subgroups, Terminology
At this time we are mainly concerned with finite groups, that is, groups with a
finite number of elements.
The order of a group, |G|, is the number of elements in the group. The order of a
group may be finite or infinite.
The order of an element, |a|, is the smallest positive integer n such that an
= e.
The order of an element may likewise be finite or infinite.
Note: if |a|=2 then a=a-1
. If |a|=1 then a=e.
A subgroup H of a group G is a subset of G together with the group operation,
such that H is also a group.
That is, H is closed under the operation, and includes inverses and identity.
(Note: H must use the same group operation as G. So Zn
, the integers mod n, is
not a subgroup of Z, the integers, because the group operation is different.)
euler portrait
http://www.math.oh
io-
state.edu/~sinnott/
ReadingClassics/h
omepage.html

15. Cancellation and
Conjugation
In any group,
a*b=a*c implies that b=c and
c*a=b*a implies that c=b.
This is used in proofs.
To conjugate an element a by x
means to multiply thus:
xax-1
or x-1
ax
While conjugating an element
may change its value, the order
|a| is preserved.
This is useful in proofs and in
solving matrix equations.
cancellation and conjugationcancellation and conjugation
http://keelynet.com/indexfeb206.htmhttp://keelynet.com/indexfeb206.htm

16. “Socks and Shoes” Property
When taking inverses of two or
more elements composed
together, the positions of the
elements reverse.
That is, (a*
b)-1
= b-1
*
a-1
. For more
elements, this generalizes to
(ab...yz)-1
= z-1
y-1
...b-1
a-1
.
In Abelian groups, it is also true
that (ab)-1
= a-1
b-1
and (ab)n
= an
bn
.
This also generalizes to more
elements.
This is called the “socks and
shoes property” as a mnemonic,
because the inverse of putting on
one's socks and shoes, in that
order, is removing ones shoes and
socks, in that order. shoes and socks in the car
http://picasaweb.google.com/mp3873/PAD#5235584405081556594
shoes and socks
http://www.inkfinger.us/my_weblog/2007/04/index.html

17. Subgroup Tests:
The One Step Subgroup Test
Suppose G is a group and H is a non-empty
subset of G.
If, whenever a and b are in H,
ab-1
is also in H,
then H is a subgroup of G.
Or, in additive notation:
If, whenever a and b are in H,
a - b is also in H,
then H is a subgroup of G.
To apply this test:
Note that H is a non-
empty subset of G.
Show that for any two
elements
a and b in H, a*b-1
is also
in H.
Conclude that H is a
subgroup of G.
Example: Show that the even integers are a subgroup
of the Integers.
Note that the even integers is not an empty set
because 2 is an even integer.
Let a and b be even integers.
Then a = 2j and b = 2k for some integers j and k.
a + (-b) = 2j + 2(-k) = 2(j-k) = an even integer
Thus a - b is an even integer
Thus the even integers are a subgroup of the integers. one step at a time by norby
http://www.flickr.com/photos/norby/37932
1413/
ab-1
H

18. Subgroup Tests:
The Two Step Subgroup Test
Let G be a group and H a
nonempty subset of G. If a●
b is in
H whenever a and b are in H, and
a-1
is in H whenever a is in H, then
H is a subgroup of G.
To Apply the Two Step
Subgroup Test:
Note that H is nonempty
Show that H is closed with
respect to the group operation
Show that H is closed with
respect to inverses.
Conclude that H is a
subgroup of G.
Example: show that 3Q*, the non-zero multiples of
3n where n is an integer, is a subgroup of Q*, the
non-zero rational numbers.
3Q* is non-empty because 3 is an element of 3Q*.
For a, b in 3Q*, a=3i and b=3j where i, j are in Q*.
Then ab=3i3j=3(3ij), an element of 3Q* (closed)
For a in 3Q*, a=3j for j an element in Q*.
Then a-1
=(j-1
*3-1
), an element of 3Q*. (inverses)
Therefore 3Q* is a subgroup of Q*.
http://www.trekearth.com/gallery/Asia/Brunei/photo653317.htm

19. Subgroup Tests:
The Finite Subgroup Test
Let H be a nonempty
finite subset of G. If H
is closed under the
group operation, then
H is a subgroup of G.
To Use the Finite Subgroup Test:
If we know that H is finite and non-empty, all we
need to do is show that H is closed under the
group operation. Then we may conclude that H
is a subgroup of G.
Example: To Show that, in Dn
, the rotations form a
subgroup of Dn
:
Note that the set of rotations is non-empty because
R0
is a rotation.
Note that the composition of two rotations is always
a rotation.
Therefore, the rotations in Dn
are a subgroup of Dn
.
math cartoons from
http://www.math.kent.edu/~sather/ugcolloq.html

20. Examples of Subgroups:
cyclic subgroups
Let G be a group, and a an element of G.
Let <a> = {an
, where n is an integer},
(that is, all powers of a.)
...Or, in additive notation,,,
let <a>={na, where n is an integer},
(that is, all multiples of a.)
Then <a> is a subgroup of G.
Note:
In multiplicative notation, a0
= 1 is
the identity; while 0a=0 is the identity in
additive notation.
Thus <a> includes the identity.
Also note that the integers less than 0 are
included here, so <a> includes all inverses.
For example:
In R*, <2>, the powers of 2,
form a subgroup of R*.
In Z, <2>, the even numbers,
form a subgroup.
In Z8
, the integers mod 8,
<2>={2,4,6,0}
is a subgroup of Z8
.
In D3
, the dihedral group of
order 6, <R120
> = {R0
, R120
, R240
}
is a subgroup of D3
Each element generates its own cyclic subgroup.
subgroup image
http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png

21. Examples of Subgroups:
The Center of a Group Z(G)
The Center of a group, written Z(G),
is the subset of elements in G which
commute with all elements of G.
If G is Abelian, then Z(G)=G.
If G is non-Abelian, then Z(G) may
consist only of the identity, or it may
have other elements as well.
For example, Z(D4
) = {R0
, R180
}.
The Center of a
Group is a
Subgroup of that
group. Subgroup lattice for D3
http://mathworld.wolfram.com/DihedralGroupD3.html

22. Examples of Subgroups:
The Centralizer of an Element
C(a)
The Centralizer of an
element C(a):
For any element a in G,
the Centralizer of a,
written C(a)
is the set
of all elements of G
which commute with a.
In an Abelian group,
C(a) is the entire group.
In a non-Abelian group,
C(a) may consist only of
the identity, a, and a-1
,
or it may include other
elements as well.
For example, in D3
,
C(f) ={f, R0
},
while C(R0
)=D3
For each element a in a group G,
C(a) = the centralizer of a is a subgroup of G.
subgroup image http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png

23. Cyclic Groups
AA Cyclic GroupCyclic Group is a groupis a group
which can be generated bywhich can be generated by
one of its elements.one of its elements.
That is, for someThat is, for some aa inin GG,,
GG={={aann
|| nn is an element ofis an element of ZZ}}
Or, in addition notation,Or, in addition notation,
GG={={nana ||nn is an element ofis an element of ZZ}}
This elementThis element aa
(which need not be unique) is called a(which need not be unique) is called a
generatorgenerator ofof GG..
Alternatively, we may writeAlternatively, we may write GG==<a><a>..
Examples:
(Z,.+) is generated by 1 or -1.
Zn
, the integers mod n
under modular addition,
is generated by 1
or by any element k in Zn
which is relatively prime to n.
Non-Examples:
Q* is not a cyclic group,
although it contains an infinite
number of cyclic subgroups.
U(8) is not a cyclic group.
Dn
is not a cyclic group although
it contains a cyclic subgroup
<R(360/n)
>
http://www.math.csusb.edu/faculty/susan/modular/modular.html

24. Properties of Cyclic Groups:
Criterion for ai
= aj
For |a| = n,
ai
= aj
iff n divides (i-j)
(alternatively, if i=j mod n.)
Or, in additive notation,
ia = ja iff i=j mod n.
For example, in Z5
,
2x4 = 7x4 = 3 because
2=7 mod 5
Corollaries:
1. |a|=|<a>| that is,
the order of an element is
equal to the order of the
cyclic group generated by
that element.
2. If ak
=e then the order of
a divides k.
For example...
in Z10
, |2|=5 and
<2>={2,4,6,8,0}
Caution: This is why it is an error to say, that the order of
an element is the power that you need to raise the
element to, to get e.
A correct statement is, the order of an element is the
smallest positive power you need to raise the element to,
to get e.
math cartoons from
http://www.math.kent.edu/~sather/ugcolloq.html

25. Properties of Cyclic Groups
For |a|=n, <ak
>=<agcd(n,k)
>
and |ak
|=n
/gcd(n,k)
For example ...
In Z30
, let a=1. Then |a|=30.
Since Z30
uses modular addition, a26
is
the same as 26. What is the order of 26?
Since gcd (26,30)=2, and gcd(2,30)=2, it
follows that |26|= |2| = gcd(2,30)=15.
Thus we expect that <26>=<2>,
and, in fact, this is {0,2,4,6,8...24,26,28}.
So we see that |<2>|=|<26>|
=30
/gcd(2,30)
= 30
/2
= 15.
In words, this reads as:
If the order of a is n,
then the cyclic group
generated by a to the k power
is the same as
the cyclic group generated by
a to the power of
the greatest common divisor
of n and k.
Also, the order of a to the k power
is equal to
the order of a
divided by
the greatest common divisor of
k and the order of a.
(Exercise: Try verbalizing a similar
statement for additive notation!)
“You may have to do something like this in a
stressful situation” - Dr. Englund

26. Properties of Cyclic Groups
For |a|=n, <ak
>=<agcd(n,k)
>
and |ak
|=n
/gcd(n,k) Corollaries....
This gives us an easy
way to specify the
generators of a group,
the generators of its
subgroups, and to tell
how these are related.
Corollary 1: When
are cyclic subgroups
equal to one another?
Let |a|=n.
Then <ai
>=<aj
> iff
gcd(n,i)=gcd(n,j)
For example ...
In Z30
, let a=1. Again, consider a2
and a26
.
Since gcd (26,30)=2, and gcd(2,30)=2,
it follows that <|26|>= <|2|> =
gcd(2,30)=15.
Thus we expect that <26>=<2>,
and, in fact, this is {0,2,4,6,8...24,26,28}.
So we see that |<2>|=|<26>|
=30
/gcd(2,30)
= 30
/2
= 15.
On the other hand, <3> ≠<2>
because gcd(30,3) = 3 while gcd (30,2)=2.
And in fact,
<3>={0,3,6...24,27} and |<3>|=30/3 = 10.
However, <3>=<9>. Do you see why?

27. Properties of Cyclic Groups
For |a|=n, <ak
>=<agcd(n,k)
>
and |ak
|=n
/gcd(n,k) Corollaries....
For example ...
In Z10
, let a=1,
so that Z10
= <a>.
Other generators for Z10
are
ak
for each k less than 10
and relatively prime to 10.
So the other generators are
3,7,and 9.
Corollary 2: Generators of
Cyclic Groups
In any cyclic group G=<a>
with order n,
the generators are ak
for each k relatively prime to n.
This gives an easy way
to find all of the
generators
of a cyclic group.
Corollary 3
specifies this for Zn
, the integers mod n under modular addition.
Since any Zn
is a cyclic group of order n,
its generators would be the positive integers less than n
and relatively prime to n.
gauss stamp
http://webpages.math.luc.
edu/~ajs/courses/322sprin
g2004/worksheets/ws5.ht
ml

28. Properties of Cyclic Groups:
The Fundamental Theorem of
Cyclic Groups
LetLet GG=<=<aa> be a cyclic> be a cyclic
group of ordergroup of order nn. Then .... Then ...
1.1. Every subgroup of aEvery subgroup of a
cyclic group is also cyclic.cyclic group is also cyclic.
2.2. The order of eachThe order of each
subgroup divides thesubgroup divides the
order of the group.order of the group.
3.3. For each divisorFor each divisor kk ofof
nn, there is, there is exactly oneexactly one
subgroup of ordersubgroup of order kk, that, that
is,is, <<aan/n/
kk
>>
For exampleFor example
considerconsider ZZ1010 = <1> with |1|=10.= <1> with |1|=10.
LetLet aa=1.=1.
Every subgroup ofEvery subgroup of ZZ1010 is also cyclic.is also cyclic.
The divisors of 10 are 1, 2, 5, and 10.The divisors of 10 are 1, 2, 5, and 10.
For each of these divisors we haveFor each of these divisors we have
exactly one subgroup ofexactly one subgroup of ZZ1010, that is,, that is,
<1>, the group itself, with order 10/1=10<1>, the group itself, with order 10/1=10
<2>={0,2,4,6,8} with order 10/2 = 5<2>={0,2,4,6,8} with order 10/2 = 5
<5>={0,5} with order 10/5 = 2<5>={0,5} with order 10/5 = 2
<10>={0} with order 10/10=1<10>={0} with order 10/10=1
The order of each of these subgroups isThe order of each of these subgroups is
a divisor of the order of the group, 10.a divisor of the order of the group, 10.
So theSo the generatorsgenerators ofof ZZ1010 would be 1, andwould be 1, and
the remaining elements: 3, 7, and 9.the remaining elements: 3, 7, and 9.
symmetry 6 ceiling art
http://architecture-buildingconstruction.blogspot.com/2006_03_01_archive.html

29. Properties of Cyclic Groups:
The Fundamental Theorem of
Cyclic Groups - Corollary
For example
consider Z10
= <1> with |1|=10.
Let a=1.
Every subgroup of Z10
is also cyclic.
The divisors of 10 are 1, 2, 5, and 10.
For each of these divisors we have
exactly one subgroup of Z10
, that is,
<1>, the group itself, with order 10/1=10
<2>={0,2,4,6,8} with order 10/2 = 5
<5>={0,5} with order 10/5 = 2
<10>={0} with order 10/10=1
The order of each of these subgroups is
a divisor of the order of the group, 10.
So the generators of Z10
would be 1, and
the remaining elements: 3, 7, and 9..
Corollary --
Subgroups of Zn
:
For each
positive divisor k of n,
the set <n
/k
>
is the unique
subgroup of Zn
of order k.
These are the
only
subgroups of Zn
.

30. Number of Elements of Each Order
in a Cyclic Group
Let G be a cyclic group of order n.
Then, if d is a positive divisor of n, then
the number of elements of order d
is φ(d) where φ is the Euler Phi function
φ(d) is defined as the number of positive
integers less than d
and relatively prime to d.
The first few values of φ(d) are:
d 1 2 3 4 5 6 7 8 9 10 11 12
φ(d) 1 1 2 2 4 2 6 4 6 4 10 4
In non-cyclic groupsIn non-cyclic groups, if, if dd is a divisor ofis a divisor of
the order of the group, thenthe order of the group, then
the number of elements of orderthe number of elements of order dd isis
aa multiplemultiple ofof φφ((dd),),
For example, consider Z12
...
Z12
= {0,1,2,3,4,5,6,7,8,9,10,11}
We have 1 element of order 2 = {6}
because φ(2)=1.
We have 2 elements of order 3 = {4,8}
because φ(3)=2.
We have 2 elements of order 4 = {3,9}
because φ(4)=2.
And 2 elements of order 6 = {2,10}
because φ(6)=2
euler totient equation
http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80
euler totient graph
http://www.123exp-math.com/t/01704079357/

31. Permutation Groups –
Definition
A Permutation of a set A
is a function
from A to A
which is both one to one
and onto
Recall from earlier work:
A function from set A to set B
is a rule which assigns
to each element of A (the domain)
exactly one element of B (the range)
One to one means f(a)=f(b)
implies that a=b.
Onto means every element of B is the
image of least one element of A under f.
Definitions
So in a Permutation on set A,
the range and the domain of the
function are both the set A.
rubic cube permutation group
http://en.wikipedia.org/wiki/Permutation_group

32. Permutation Groups –
Definitions
Definitions:
A Permutation Group of a set A
is a set of Permutations on A which form
a group under function composition.
That is, the elements are functions
from A to A
which are both one to one
and onto, and the binary operation is
function composition,
Recall from earlier work:
A group has elements which include
identity and inverses, and an operation
which is associative and closed.
So here, the elements are functions and
the operation is function composition,
which is always associative (although not
usually commutative.)
This may take some “getting used to”,
because in most of the groups we have
seen so far, the elements are values and
the operation is a function.

33. Permutation Groups -- Discussion
So in a Permutation Group, we need
functions and their inverses; also we
need an identity function.
Recall from earlier work that
a function has an inverse
if and only if
it is one to one and onto.
Since functions are group elements,
we expect that
a function composed with its inverse
will result in the identity function, as
indeed it does.
At this time we are mainly
concerned with permutations on a
finite set, so we can build closure
into the function definitions.

34. Permutation Groups –
Notations
In other areas such as algebra and
calculus, functions are defined on
infinite sets and often written as
algebraic formulas.
However, in this context we usually
define a function explicitly, by listing
an element of the domain along side
its corresponding function value.
There are two ways this is
commonly done: grid notation (due
to Euler) and cycle notation (due to
Cauchy).
Notations
math cartoons from
http://www.math.kent.edu/~sather/ugcolloq.html

35. Permutation Groups –
Notations
Consider D4, the symmetry group of a
square. We can represent this group as
a permutation group in grid notation as
follows.
Number the four corners of the square:
1,2,3,4 as shown.
Then we can represent a 90 o
counterclockwise rotation as:
R90 =
1 2 3 4
2 3 4 1
and a reflection across the horizontal
axis as:
Fh
= 1 2 3 4
2 1 4 3
Example, D4, the symmetries of a Square
Or, in cycle notation we can write:
R90
= (1 2 3 4)
and
Fh
= (1 2)(3 4)
1
23
4

36. Permutation Groups –
Definitions and Notation
Let S3
be the group of all one to one
functions from the set {1, 2, 3} to itself.
This is the same as saying, all
arrangements or permutations of these
three elements.
S3
and Sn
Recall from previous work that the
number of permutations on a set of
n elements is n! So the number of
elements in S3
is 3! or 6.
In general, Sn
is the set of all
permutations on a set of size n, and
the order of Sn
is n!
subgroup lattice for s3
http://www.mathhelpforum.com/math-help/advanced-algebra/22850-normal-
subgroup.html

37. Cauchy Cycle Notation
Disjoint Cycles Commute
Order of a Permutation
Cycle notation, introduced by Cauchy,
has many advantages over grid notation.
In general, the product of cycles does
not commute. However, we can write any
permutation as a cycle or a product of
disjoint cycles, that is, each element of
the set appears in at most one cycle.
Written in this form,
disjoint cycles commute.
Also when we write a permutation in this
form,
the order of the permutation
(that is, the smallest number of times we
need to repeat that permutation to obtain
the identity permutation) is
the least common multiple
of the lengths of the cycles.

38. Any permutation may also be written or
“decomposed” as a product of two-cycles.
This product would usually not be disjoint,
and it need not be unique.
However, if a permutation can be written as
an odd number of two-cycles, any other
decomposition of that permutation is also an
odd number of two cycles.
Similarly, if a permutation can be written as
an even number of two-cycles, any other
decomposition of that permutation is also an
even number of two cycles.
The identity permutation is always an
even number of two-cycles.
Decomposition into two-
cycles
Always even or always odd
Identity is always even
Cycle
Notation
“I’m odd and I’ll always be odd!-- Joni This oddfellow is still odd even though he’s decomposed!
IOOF tombstone
http://farm1.static.flickr.com/203/485879582_fa65f4e4b1.jpg?v=0

39. Cycle
Notation
If a permutation can be
written as an odd number of
two cycles, we say it is an
odd permutation.
If a permutation can be
written as an even number of
two cycles, we say it is an
even permutation.
Definitions, Examples
Even Permutation
Odd Permutation
Examples of decomposition into 2-
cycles:
(12345) = (15)(14)(13)(12)
(1234) = (14)(13)(12)
(1234)(247)=(14)(13)(12)(27)(24)
Notice that (12345) is a cycle of
length 5, and 5 is an odd number –
yet (12345) can be written as an
even number of two-cycles so it is
an even permutation.
However, (1234), a cycle of length 4
is an odd permutation because it
can be written as an odd number of
two-cycles, even though 4 is an
even number.
“A cycle of odd length is always even –
now that's odd! “ -- Rus

40. The Alternating Groups
of Degree n
For any n,
the set of even permutations in
Sn
forms a group, and a
subgroup of Sn.
This is because the inverse of an
even permutation is always even,
the composition of even cycles is
always even, and the identity can
always be written as an even cycle.
This group is called An
, the
alternating group of degree n. When
n ≠1, An
is of order n!
/2
The Alternating groups are very
important in historical, theoretical,
and applications contexts.
An
permutation group A4
http://faculty.smcm.edu/sgoldstine/origami/display
text.html
Note that the set of odd
permutations do not form a
group, because this set lacks
the identity permutation and
lacks closure under
composition.
Permutation group A5
http://www.math.metu.edu.tr/~berkman/466object.html

41. Suggested Activities
Cycle Notation,
more practice
Cycle notation,
introduced by Cauchy,
has many advantages
over Euler's grid
notation, although some
people may require more
practice to perform
calculations quickly and
accurately in this
notation.
For practice, try verifying
the Cauchy table for A4
on page 105 of the text,
by multiplying out each
pair of elements and
showing that the product
is as stated in the table.

42. Suggested Activities
Reinforcement of Knowledge
Base
Modern educational practices tend to
de-emphasize knowledge base in
favor of constructivist education
theory, which is thought to better
support higher order thinking.
However, for most people*, an
extensive knowledge base is
necessary for mastery of this subject.
Please see the textbook author
Gallian’s comments this matter, on
his website
http://www.d.umn.edu/~jgallianhttp://www.d.umn.edu/~jgallian under
the heading “Advice for students for
learning abstract algebra”
*Sylvester notwithstanding, text pg 89
One who has the facts memorized
with understanding is comparable to
a highly skilled worker who has the
proper tools on hand and knows how
to use them, rather than having to
scrounge around and improvise with
a stick or a rock!Tools 3 www.istockphoto.com
tools and rocks www.dkimages.com

43. Often students complain that memorizing facts
and definitions is excessively difficult, or that
knowledge obtained this way does not seem to
apply to testing or problem solving situations.
Complaints like this often result from failure to
adapt study habits to one's particular learning
style. For example, a visual learner may get by
with simply reading over text or notes, while an
aural learner may need to say lessons out loud or
even set them to music. A kinesthetic learner
often needs to re-write the lessons, make models,
demonstrate the information to others, etc.
Although kinesthetic methods are more demanding
in terms of time and energy, all learners benefit
from kinesthetic learning methods, because these
methods produce more robust memory traces in the
brain.
Suggested Activities:
Reinforcement of Knowledge Base according to
individual learning style

44. Suggested Activities:
Reinforcement of Knowledge Base
according to individual learning style
This is a sample Memletics Learning Styles graph produced by the online test at
http://www.learning-styles-online.com/inventory/http://www.learning-styles-online.com/inventory/ Picture credit http://www.learning-styles-online.com/inventory/Picture credit http://www.learning-styles-online.com/inventory/
Please visit one or both of these sites to
find out or confirm your learning style:
http://www.usd.edu/trio/tut/ts/stylestest.htmlhttp://www.usd.edu/trio/tut/ts/stylestest.html
for a brief quiz and classification into one of
three styles, or
http://www.learning-styles-online.comhttp://www.learning-styles-online.com
for a more extensive questionnaire and
classification into one of six styles. This site
will generate a detailed description of your
learning style and a graph showing your
style according to six axes.
Both sites have been checked for safety
from malware, and contain
recommendations and resources for the
various learning styles.

45. Suggested Activities:
Online Resources provided by
Text Author J.A. Gallian
http://www.d.umn.edu/~jgallian
Practice with the true-false questions and flashcards
provided on this website may be particularly helpful for
building and reinforcing knowledge base. The
flashcards may be printed out or used in software
form, from the website.
This website also contains links to some useful
software. In particular, the Group Explorer
software, available on the above website by
following the link Group Explorer, or one may go
directly to:
http://groupexplorer.sourceforge.net/
quarternion group from
http://home.att.net/~numericana/http://home.att.net/~numericana/
answer/groups.htmanswer/groups.htm
rock stack from loveringllc.com

46. Suggested Activities:
Other Online Resources
http://en.wikipedia.org/wiki/Group_theory
(a good starting point)
http://www.math.miami.edu/~ec/book/
(a free online book on abstract algebra)
http://en.wikipedia.org/wiki/Dihedral_group
(explains the dihedral groups)
http://members.tripod.com/~dogschool/cyclic.html
(explains Cauchy cycle notation)
http://en.wikipedia.org/wiki/Euler%27s_totient_function
(Totient function, in Gallian text
referenced in connection with U-groups)
http://members.tripod.com/~dogschool/index.html
(an index to another online text)
http://en.wikipedia.org/wiki/Symmetric_group
(explanations and applications of symmetric groups)
http://www.t209.com/article.php?art_id=26
(a vbasic program φ(n) for numbers up to 231
)
http://eldar.mathstat.uoguelph.ca/dashlock/math3130/pdf/Chapter2b.pdf
(another explanation of Cayley cycle notation)
I
Here are a few other good online resources. All of these have been
checked for malware and are safe.
http://www.mathwire.com/seasonal/winter07.html

47. Acknowledgments
Dr. T. Englund, Professor of Mathematics, Central Washington University,
professor for Mathematics 461
Dr. T. Willard, Professor of Mathematics, Central Washington University,
graduate adviser
I am grateful for the cooperation of the following fourth year undergraduate
Mathematics students at Central Washington University, who included me in their
study group for the final exam. Notes from this study group were then composed
into this presentation:
Amber Goodrich, Mike Prothman, Russel Hess,
David Melik, Geoff LaBrandt,
Brandon Belieu,
Special thanks to:
Two Bears, my brother St. Euler St. Cauchy
All graphics in this presentation not created by J. Pinkney are reproduced from online sources
according to the Fair Use Provisions of the U.S. Copyright Act, with the html sources cited in the frame
on which they appear.