Lecture 14sd

8/12/2019 Lecture 14sd

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Let Sbe a non-empty set.Let!be a binary operator on S.

Definition of a Group

(S,!) is called a groupif it has these properties:

,a b S a b S ! " !closure:

associativity: , , ( ) ( )a b c S a b c a b c! " =

identity: s.t.e S a S e a a e a! " # " = =

inverse: 1 1 1s.t.a S a S a a a a e! ! !" # $ # = =


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Examples of (abstract) groups

(the integers) is a group under operation +

(the reals) is a group under operation +

+(the positive reals) is a group under !

\ {0} is a group under !

Zn(the integers mod n) is a group under +

Zn*is a group under !


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NONEXAMPLES of groups

, operation "

\ {0}, operation !

G = {all odd integers}, operation +

+ is not a binary operation on G!

"is not associative!

1 is the only possible identity element;

but then most elements dont have inverses!


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A permutation!on [1..n]is a one-to-one

function [1..n]mapping onto [1..n].

1 1 ( ) ( )

onto [1.. ] s.t. ( )

x y x y

y n x x y

! !

!

" # $ #

% & ' =

1 12 2

3 34 4

1 2 3 4

1 3 2 4

! "# $% &


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Composition of permutations

Let!1and!2be permutations on [1..n].

The composition of!1and!2, written

!2 !1is given by

!2 !1 (x) =!2(!1 (x) )

x

!1 !2

!2(!1 (x) )


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Sn= set of all permutations on [1..n]!= permutation composition

Theorem: (Sn,!) is a group

Proof:

closurec b a

associativity identitya

-

a-1

inverse

a!b!ca!b

b a


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x x

x

Commutative caution:Permutation composition doesnt always commute.

123

312

! "# $# $# $% &

123

231

! "# $# $# $% &

123

132

! "# $# $# $% &

123

213

! "# $

# $# $% &123

132

! "

# $# $# $% &

123

213

! "

# $# $# $% &

x


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x x

x

3 books on a shelf

123

312

! "# $# $# $% &

123

231

! "# $# $# $% &

x

Swap book 1and book 2

Swap book 2

and book 3


Swap book 1

and book 2


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Cycle Representation of a Permutation

A cycle (a1a2 ak) is a permutation that sends a1to a2,a2to a3, , ak-1to ak, akto a1, and leaves the rest ofthe elements unchanged.

Every permutation can be represented as thecomposition of disjoint cycles:

12345678

24316587

! "# $% &

= (1 2 4)(3)(5 6)(7 8)


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x x

x

Cycle representation

x

Cycle book 1and book 2

Cycle book 2

and book 3


Swap book 1

and book 2

(2 3) (1 2) (2 3) (1 2)


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Let Sbe a non-empty set.Let!be a binary operator on S.

Definition of a Group

(S,!) is called a groupif it has these properties:

,a b S a b S ! " !closure:

associativity: , , ( ) ( )a b c S a b c a b c! " =

identity: s.t.e S a S e a a e a! " # " = =

inverse: 1 1 1s.t.a S a S a a a a e! ! !" # $ # = =


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Proof:

Cancellation Theorem


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Z6= {0, 1,2,3,4,5} with ++ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 1 2 3 4 5 0

2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

An operator hasthe permutation

property ifeach row and

each column hasa permutation

of theelements.


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Z5*

= {1,2,3,4}

*5 1 2 3 41 1 2 3 4

2 2 4 1 3

3 3 1 4 24 4 3 2 1


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The column permutation property is

equivalent to the right cancellationproperty:

[b * a = c * a] )b=c

* 1 2 a 4

b 1 2 3 4

2 2 4 1 3

c 3 1 4 2

4 4 3 2 1


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The row permutation property is

equivalent to the left cancellationproperty:

[a * b = a *c] )b=c

* b 2 c 4

1 1 2 3 4

2 2 4 1 3

a 3 1 4 2

4 4 3 2 1


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Row2(a) = 2 * a

*5 1 2 3 41 1 2 3 4

2 2 4 1 3

3 3 1 4 24 4 3 2 1


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Cayleys Theorem:The * table for any group (G,*) has thepermutation property.

8g2G define Rowa(g) = a*g

By cancellation, x"y )Rowa(x) "Rowa(y)Rowais a 1-1 function from G to G.,

Pick any g2G. Rowa(a-1*g )= g, so Rowa Iis ONTO.

Rowais a permutation of G.Rowa*b(x) = Rowa( Rowb (x) )

can be seen as the row permutations undercomposition.


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Order of a group element

Let G be a finite group. Let a#G.

Look at 1, a, a2, a3, # till you get some repeat.

Say ak= ajfor some k > j.

Multiply this equation by a"j

to get ak"j

= 1.

So the first repeat is always 1.

Definition: The order of x, denoted |a|, is the

smallest m !1 such that am= 1.

Note that a, a2, a3, #, am"1, am=1 all distinct.


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Order Theorem:

|a| always divides evenly into |G|.

G1a

a2

a3am"1

xxa

xa2

xa3xam"1

Claim: also of length m.

Because xaj= xak $ aj= ak.


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Order Theorem:


G1a

a2

a3am"1

xxa

xa2

xa3xam"1

yya

ya2Impossible.

Multiply on right by a"1

.


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Order Theorem:


G1a

a2

a3am"1

xxa

xa2

xa3xam"1

yya

ya2ya3

yam"1

G partitioned

into cycles ofsize m.


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Order Theorem:


Corollary: If |G| = n, then an=1 for all a#G.

Proof: Let |a| = m. Write n = mk.

Then an= (am)k= 1k= 1.

Corollary: Eulers Theorem: a$(n)= 1 in Zn*


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15-251: Great Theoretical Ideas in Computer Science

Group Theory


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Group Theory

Study of symmetries and transformations

of mathematical objects.

Also, the study of abstract algebraic

objects called groups.


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What is group theory good for?

Checksums, error-correction schemes

Minimizing space-complexity of algorithmsMinimizing randomness-complexity of algorithms

Cryptosystems

Algorithms for quantum computers

Hard instances of optimization problems

Ketan Mulmuleys approach to P vs. NP

In theoretical computer science:


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15 Puzzle

Rubiks Cube

SET

Tangles

In puzzles and games:


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Theres a quadratic formula:

In math:


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Theres a cubic formula:

In math:


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Theres a quartic formula:

In math:


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There is NOquintic formula.

In math:


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Predicting the existence of elementary

particles before they are discovered.

In physics:


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Driving the plot of S06E10 of Futurama,

The Prisoner of Benda

In entertainment:


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So: What is group theory?


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Rotate


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Flip


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Head-to-Toe flip


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Q: How many positions can it be in?

A: Four.


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1 2

43

4 3

12

2 1

34

3 4

21

Rotate

Flip

Head-

to-Toe

Flip

Rotate


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Group theory is not so much

about objects (like mattresses).

Its about the transformations

on objects and how they (inter)act.


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1 2

43

4 3

12

2 1

34

3 4

21

R

F

H

F

R

F(R(mattress)) =

H(mattress)

H(F(mattress)) =

R(mattress)

R(F(H(mattress))) =

mattressId(mattress)

FR=H

HF=R

RFH=Id

RIdHFH= H


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The kinds of questions asked:

What is RIdHFH ?

Do transformations Aand Bcommute?

I.e., does AB = BA ?

What is the order of transformation A?

I.e., how many times do you have to

apply Abefore you get to Id?


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Definition of a group of transformations

Let X be a set.

Let G be a set of bijections p : X %X.

We say G is a groupof transformations if:

1. If pand qare in G then so is pq.

G is closedunder composition.

2. The do-nothing bijection Idis in G.

3. If pis in G then so is its inverse, p!1.

G is closed under inverses.


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Example: Rotations of a rectangular mattress

X = set of all physical points of the mattress

G = { Id, Rotate, Flip, Head-to-toe }

Check the 3 conditions:



3. If pis in G then so is its inverse, p!1

.

!

!

!


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Example: Symmetries of a directed cycle

X = labelings of the

vertices by 1,2,3,4

2

3

1

4

|X| = 24

G = permutations

of the labels which

dont change the graph

|G| = 4

G = { Id, Rot90, Rot180, Rot270}


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vertices by 1,2,3,4

|X| = 24

G = permutations

of the labels which


|G| = 4

G = { Id, Rot90, Rot180, Rot270}


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G = { Id, Rot90, Rot180, Rot270}

X = labelings of directed 4-cycle

Check the 3 conditions:



3. If pis in G then so is its inverse, p!1.

!

!

!

Cyclic group of size 4


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Example: Symmetries of undirected n-cycle


vertices by 1,2, #, n

|X| = n!

G = permutationsof the labels which


|G| = 2n

1

2

34

5


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|X| = n!



|G| = 2n

2

1

54

3

+ one clockwise twist


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|X| = n!



|G| = 2n

3

2

15

4

+ one clockwise twist =


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|X| = n!



|G| = 2n

G = { Id, n"1 rotations, n reflections}

Dihedral group of size 2n


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Example: All permutations

X = {1, 2, #, n}

G = all permutations of X

e.g., for n = 4, a typical element of G is:

Symmetric group, Sym(n)


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More groups of transformations

Motions of 3D space: translations + rotations

(preserve laws of Newtonian mechanics)

Translations of 2D space by an integer amounthorizontally and an integer amount vertically

Rotations which preserve an

old-school soccer ball.

|G| = 60


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Group theory is not so much

about objects (like mattresses).

Its about the transformations

on objects and how they (inter)act.


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1 2

43

4 3

12

2 1

34

3 4

21

R

F

H

F

R

F(R(mattress)) =

H(mattress)

H(F(mattress)) =

R(mattress)

R(F(H(mattress))) =

mattressId(mattress)

FR=H

HF=R

RFH=Id

RIdHFH= H

There is no mattress.


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The laws of mattress rotation

G = { Id, R, F, H }

Id Id = Id

Id R = R

Id F = FId H = H

R Id = R

R R = Id

R

F = H

R H = F

F Id = F

F R = H

F F = IdF H = R

H Id = H

H R = F

H

F = RH H = Id


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The laws of the dihedral group of size 10

G =

{ Id, r1, r2, r3, r4,f1, f2, f3, f4, f5 }

Id r1 r2 r3 r4 f1 f2 f3 f4 f5

Id Id r1 r2 r3 r4 f1 f2 f3 f4 f5

r1 r1 r2 r3 r4 Id f4 f5 f1 f2 f3

r2 r2 r3 r4 Id r1 f2 f3 f4 f5 f1

r3 r3 r4 Id r1 r2 f5 f1 f2 f3 f4

r4 r4 Id r1 r2 r3 f3 f4 f5 f1 f2

f1 f1 f3 f5 f2 f4 Id r3 r1 r4 r2

f2 f2 f4 f1 f3 f5 r2 Id r3 r1 r4

f3 f3 f5 f2 f4 f1 r4 r2 Id r3 r1

f4 f4 f1 f3 f5 f2 r1 r4 r2 Id r3

f5 f5 f2 f4 f1 f3 r3 r1 r4 r2 Id


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Let G be a set.

Let be a binary operation on G;

think of it as defining a multiplication table.

a b c

a c a b

b a b cc b c a

E.g., if G = { a, b, c } then#

#is a binary operation.

This means that ca = b.

Lets define an abstract group.


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Definition of an (abstract) group

We say G is a groupunder operation if:

1. Operation is associative:

i.e., a(bc) = (ab)c %a,b,c#G

2. There exists an element e#G

(called the identity element) such that

ae = a, ea = a %a#G

3. For each a#G there is an element a"1#G

(called the inverse of a) such that

aa"1= e, a"1a = e


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Any group of transformations is a group.

(Only need to check that composition of functions is associative.)

E.g., the mattress group (AKA Klein 4-group)

Id R F H

Id Id R F H

R R Id H F

F F H Id R

H H F R Id

identity element is Id

R"1 = R

F"1 = F

H"1 = H


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Check:

0. + really is a binary operation on 1. + is associative: a+(b+c) = (a+b)+c

2. e is 0: a+0 = a, 0+a = a

3. a"1 is "a: a+("a) = 0, ("a)+a = 0


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(the reals) is a group under operation +

+(the positive reals) is a group under !

\ {0} is a group under !

Zn(the integers mod n) is a group under +

Zn*is a group under !


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NONEXAMPLES of groups

, operation "

\ {0}, operation !

G = {all odd integers}, operation +

+ is not a binary operation on G!

"is not associative!

1 is the only possible identity element;but then most elements dont have inverses!


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Abstract algebra on groups

Theorem 1:If (G,) is a group, identity element is unique.

Proof:

Suppose f and g are both identity elements.Since g is identity, fg = f.

Since f is identity, fg = g.

Therefore f = g.


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Theorem 2:In any group (G,), inverses are unique.

Proof:

Given a#G, suppose b, c are both inverses of a.Let e be the identity element.

By assumption, ab = e and ca = e.

Now: c = ce= c(ab)

= (ca)b = eb = b


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Theorem 3:For all a in group G we have (a"1)"1= a.

Theorem 4:

For a,b#

G we have (ab)"1

= b"1

a"1

.

Theorem 5:

In group (G,), it doesnt matter how you

put parentheses in an expression likea1a2a3&&&ak

(generalized associativity).


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Notation

In abstract groups, its tiring to always write .

So we often write ab rather than ab.

For n#+, write aninstead of aaa&&&a (n times).

Also a"ninstead of a"1a"1&&&a"1, and a0means 1.

Sometimes write 1 instead of e for the identity.


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Algebra practice

Problem: In the mattress group {1, R, F, H},simplify the element R2 (H3 R"1)"1

One (slightly roundabout)solution:

H3 = H H2= H 1 = H, so we reach R2 (HR"1)"1.

(HR"1)"1= (R"1)"1 H"1= R H, so we get R2R H.

But R2= 1, so we get 1 R H = R H = F.

Moral: the usual rules of multiplication, except#


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Commutativity?

In a group we do NOT NECESSARILY have

a b = b a

Actually, in the mattress group we do havethis for all elements. E.g., RF = FR (=H).

Definition:

a,b#G commute means ab = ba.G is commutative means all pairs commute.


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In group theory, commutative groups

are usually called abelian groups.

Niels Henrik Abel (1802"1829)

NorwegianDied at 26 of tuberculosis !

Age 22: proved there is

no quintic formula.


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Evariste Galois (1811"1832)

FrenchDied at 20 in a dual !

One of the main inventors

of group theory.


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Some abelian groups:

Mattress group (Klein 4-group)

Symms of a directed cycle (cyclic group)

(, +), (Zn*,!)

Some nonabelian groups:

Symms of an undirectedcycle (dihedral group)

Motions of 3D space

Sym(n) (symmetric group on n elements)


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More fun groups:

Quaternion group

Q8= { 1, "1, i, "i, j, "j, k, "k }

Multiplication 1 is the identity

defined by: ("1)2

= 1, ("1)a = a("1) = "ai2= j2= k2= "1

ij = k, ji = "k

jk = i, kj = "i

ki = j, ik = "j

Exercise: valid def. of a (nonabelian) group.


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Application to computer graphics

Quaternions: expressions like

3.2 + 1.4i ".5j +1.1k

which generalize complex numbers ().

Let (x,y,z) be a unit vector, 'an angle, let

q = cos('/2) + sin('/2)x i + sin('/2)y j + sin('/2)z k

If P is a point 3D space, then qPq"1is its

rotation by angle 'around axis (x,y,z).


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More fun groups:

Matrix groups

SL2(m): Set of matrices

where a,b,c,d#Zm and ad"bc=1.

Operation: matrix mult. Inverses:

Application: constructing expander graphs,

magical graphs crucial for derandomization.

I hi


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Isomorphism

Id R F H

Id Id R F H

R R Id H F

F F H Id R

H H F R Id

Heres a group: V = { (0,0), (0,1), (1,0), (1,1) }

+ modulo 2 is the operation

Theres something familiar about this group!

+ 00 01 10 11

00 00 01 10 11

01 01 00 11 10

10 10 11 00 01

11 11 10 01 00

The mattressV sameafter

renaming:

00(Id

01(R

10(F

11(H

I hi


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Isomorphism

Groups (G,) and (H,) are isomorphic

if there is a way to rename elements so that

they have the same multiplication table.

Fundamentally,

theyre the same abstract group.

I hi d d


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Isomorphism and orders

Obviously, if G and H are isomorphic

we must have |G| = |H|.

|G| is called the orderof G.

E.g.: Let C4be the group of transformations

preserving the directed 4-cycle.

|C4| = 4

Q: Is C4isomorphic to the mattress group V ?

I hi d d


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Isomorphism and orders

Q: Is C4isomorphic to the mattress group V ?

A: No!

a2= 1 for every element a#V.

But in C4, Rot902 = Rot270

2 " Rot1802= Id2

Motivates studying powers of elements.


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Order of a group element

Let G be a finite group. Let a#G.

Look at 1, a, a2, a3, # till you get some repeat.

Say ak= ajfor some k > j.

Multiply this equation by a"jto get ak"j= 1.

So the first repeat is always 1.

Definition: The order of x, denoted |a|, is the

smallest m !1 such that am= 1.

Note that a, a2, a3, #, am"1, am=1 all distinct.


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Examples:

In mattress group (order 4),|Id| = 1, |R| = |F| = |H| = 2.

In directed-4-cycle group (order 4),

|Id| = 1, |Rot180| = 2, |Rot90| = |Rot270| = 4.

In dihedral group of order 10

(symmetries of undirected 5-cycle)

|Id| = 1, |any rotation| = 5, |any reflection| = 2.


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Order Theorem:


G1a

a2

a3am"1

xxa

xa2

xa3xam"1

Claim: also of length m.

Because xaj= xak $ aj= ak.


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Order Theorem:


G1a

a2

a3am"1

xxa

xa2

xa3xam"1

yya

ya2Impossible.

Multiply on right by a"1.


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Order Theorem:


G1a

a2

a3am"1

xxa

xa2

xa3xam"1

yya

ya2ya3

yam"1

G partitioned

into cycles ofsize m.


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Order Theorem:


Corollary: If |G| = n, then an=1 for all a#G.

Proof: Let |a| = m. Write n = mk.

Then an= (am)k= 1k= 1.

Corollary: Eulers Theorem: a$(n)= 1 in Zn*


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Check Digits

Say you have important strings of digits:

credit card numbers

EFT routing numbers

UPC numbersmoney serial numbers

book ISBNs

People screw up when transcribing them.


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Check Digits

Commonest human screwups:

single digit wrong (e.g., 6%8): 60-90%

omitting/adding digit: 10-20%

transposition (e.g., 35%53): 10-20%other screwups: " 5%

Instead of making them n random digits,make them n random digits + a check digit.


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Check Digits

Example: Book ISBNs before 2007.

Desired id#: 1 3 6 0 4 2 9 9 4

10 9 8 7 6 5 4 3 2

dot-prod mod 11: 1!10+3!9+6!8+0!7+4!6+2!5+9!4+9!3+8!2= 4

check digit: top it off to get 0 mod 11

7

Pros: You can detect any single-digit or

transposition error.


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Check Digits

Example: Book ISBNs before 2007.

Desired id#: 1 3 6 0 4 2 9 9 4

10 9 8 7 6 5 4 3 2

dot-prod mod 11: 1!10+3!9+6!8+0!7+4!6+2!5+9!4+9!3+8!2= 4

check digit: top it off to get 0 mod 11

7

Cons: Um, check digit should be 10? Write X!

Doesnt scale if you want longer id#s.


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Verhoeff Check Digit Method

Encode digits by elements of dihedral group of order 10.

Let )be the permutation

Given a desired id# a0a1a2&&&an"1,choose unique check digit ansatisfying group equation

enc()0(a0))enc()1(a1))enc()

2(a2))&&&enc()2(an)) = e

Pros: Detects single-digit & transposition errors.Uses just digits 0, 1, 2, #, 9.

Scales to any length of id#.


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2(a2))&&&enc()2(an)) = e

Cons: Cant really be done by a human.


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2(a2))&&&enc()2(an)) = e

Wait, is this even a con?What human manually checksums credit cards?

We have computers, you know.


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Nevertheless, its like the Dvorak keyboard

of check digit methods. !

German federal bank started

using it for Deutsche Marks

(with some letters?) in 1990.

Then they went and got the euro(which uses a different scheme).


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The 10 is a good denomination

for mathematicians.

Leonhard Euler on the back of the old

10 Swiss franc note.


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The 10 is a good denomination

for mathematicians.

Cahit Arf and an equation in the group Z2

starring on the back of a Turkish 10 lira.

I do not know

how this works.

Definitions:


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Definitions:

Groups

Commutative/abelian

Isomorphism

Order

Groups:

Klein 4-, cyclic, dihedral,symmetric, quaternions

Doing:

Checking for groupness

Computations in groups

Theorem/proof:

Study Guide

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Lecture 14sd