A GROUP has the following properties:

A GROUP has the following properties:

• Closure

• Associativity

• Identity

• every element has an Inverse

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G = { i, k, m, p, r, s } is a group with operation *as defined below:

G has CLOSURE:for all x and y in G,x*y is in G.

The IDENTITY is i :for all x in G, ix = xi = x

Every element in G has an INVERSE:k*m = ip*p = ir*r = is*s = i

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G has ASSOCIATIVITY:for every x, y, and z in G,(x*y)*z = x*(y*z) for example:

( k*p )* r( s )* r

m

= k* ( p* r ) k* ( k )

m

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G = { i, k, m, p, r, s } is a group with operation *as defined below:

G does NOT haveCOMMUTATIVITY:

p*r = r*p

H = { i, k, m }is a SUBGROUP

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definition: If G is a group, H is a subgroup of G, and g is a member of G then we define the left coset gH:

gH = { gh / h is a member of of H }


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Example: to form the coset r H


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H = { i , k , m }r r r r


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H = { i , k , m }r r r r

= { r , s , p } s p r

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srpmki*H = { i, k, m } = a subgroup

The COSETS of H are:

iH = { i*i, i*k, i*m }={i,k,m}

kH = { k*i, k*k, k*m }={k,m,i}

mH = {m*i,m*k, m*m}={m,i,k}

pH = { p*i, p*k, p*m }={p,r,s}

rH = { r*i, r*k, r*m }={r,s,p}

sH = { s*i, s*k, s*m }={s,p,r}

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srpmki*The cosets of a subgroupform a group:

A BA A BB B A

EFGHCDABH

ACHGFBEDG

BDEFGAHCF

HGFEDCBAE

CABDEHFGD

DBACHEGFC

FEDBAGCHB

GHCABFDEA

HGFEDCBA#

M = { A,B,C,D,E,F,G,H } is a noncommutative group.

N = { B, C, E, G } is a subgroup of M

EFGHCDABH

ACHGFBEDG

BDEFGAHCF

HGFEDCBAE

CABDEHFGD

DBACHEGFC

FEDBAGCHB

GHCABFDEA

HGFEDCBA#

EFGHCDABH

ACHGFBEDG

BDEFGAHCF

HGFEDCBAE

CABDEHFGD

DBACHEGFC

FEDBAGCHB

GHCABFDEA

HGFEDCBA#

The cosets of N = { B, C, E, G } are:

EFGHCDABH

ACHGFBEDG

BDEFGAHCF

HGFEDCBAE

CABDEHFGD

DBACHEGFC

FEDBAGCHB

GHCABFDEA

HGFEDCBA#

AN = { D,F,A,H }

DN = { F,H,D,A }

FN = { H,A,F,D }

HN = { A,D,H,F }

BN = { C,G,B,E }

CN = { G,E,C,B }

EN = { B,C,E,G }

GN = { E,B,G,C }

EGCBFHDAH

BEGCDFAHF

CBEGADHFD

GCBEHAFDA

AHFDCGBEG

HFDAGECBE

DAHFBCEGC

FDAHEBGCB

HFDAGECB#

Rearrange the elements of the table so that members or each cosetare adjacent and see the pattern!

ihgfedcbai

hgfedcbaih

gfedcbaihg

fedcbaihgf

edcbaihgfe

dcbaihgfed

cbaihgfedc

baihgfedcb

aihgfedcba

ihgfedcba&

Q is a commutative groupR = { c, f, I } is a subgroup of Q

ihgfedcbai

hgfedcbaih

gfedcbaihg

fedcbaihgf

edcbaihgfe

dcbaihgfed

cbaihgfedc

baihgfedcb

aihgfedcba

ihgfedcba& The cosets of R:

{ d,g,a }

{ e,h,b,}

{ c,f,I }

a b c

d e f

g h i

The cosets of a subgroup partition the group:

LAGRANGE’S THEOREM: the order of a subgroup is a factor of the order of the group.

ie: every member of the group belongs to exactly one coset.

(The “order” of a group is the number of elements in the group.)

ebhfcigdag

bhecifdagd

hebifcagda

fcigdahebh

cifdagebhe

ifcagdbheb

gdahebifci

dagebhfcif

agdbhecifc

gdahebifc&

If we rearrange the members of Q, we can see that the cosets form a group

ebhfcigdag

bhecifdagd

hebifcagda

fcigdahebh

cifdagebhe

ifcagdbheb

gdahebifci

dagebhfcif

agdbhecifc

gdahebifc&

example 1: the INTEGERS with the operation +

closure: the sum of any two integers is an integer.

associativity: ( a + b ) + c = a + ( b + c )

identity: 0 is the identity

every integer x has an inverse -x

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

The multiples of three form a subgroup of the integers:

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

With coset: (add 1 to every member of T)

Z

T

example 1: the INTEGERS with the operation +

closure: the sum of any two integers is an integer.

associativity: ( a + b ) + c = a + ( b + c )

identity: 0 is the identity

every integer x has an inverse -x

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

The multiples of three form a subgroup of the integers:

{………-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …………}

With coset: (add 1 to every member of T)

Z

T

and coset (add 2 to every member of T)

example 2: The set of all points on the plane with operation +defined: The identity is the origin.

db

ca

d

c

b

a

R2 =

example 3: The set of points on a line through the origin is a SUBGROUP of R2. eg: y = 2x

If the vector is added to every point on y = 2x

1

2

You get a coset of L

L=

Theorem: Every group has the cancellation property.

No element is repeated in thesame row of the table. No element is repeated in the same column of the table.



rra

yx

Because r is repeated in the row,

if a x = a y you cannot assume that x = y .

In other words, you could not “cancel” the “a’s”



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then

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11

In a group, every element has an inverse and you have associativity.

r s t u v w

r s

s t r

t

u t

v s

w v r

COMPLETE THE TABLE TO MAKE A GROUP:

r s t u v w

r s

s t r

t

u t

v s

w v r

What is the IDENTITY?

If r were the identity, then rw would be w

If s were the identity, then sv would be v

If w were the identity, then wr would be r

r s t u v w

r s

s t r

t

u t

v s

w v r

The IDENTITY is t

r

tr = r

r s t u v w

r s

s t r

t

u t

v s

w v r

The IDENTITY is t

r

tr = r

ts = s

s

r s t u v w

r s

s t r

t

u t

v s

w v r

The IDENTITY is t

r

tr = r

ts = s

s

tt = t

ttu = u

utv = v

v

tw = w

w

r s t u v w

r s

s t r

t

u t

v s

w v r

The IDENTITY is t

r s

tt = t

t u v w

and

rt = r

r

st = s

sut = u

uvt = v

vwt = w

w

r s t u v w

r r s

s s t r

t r s t u v w

u u t

v v s

w v w r

sv = t

s and v are INVERSES

vs = t

t

r s t u v w

r r s

s s t r

t r s t u v w

u u t

v t v s

w v w r

u is its own inverse

r s t u v w

r r s

s s t r

t r s t u v w

u u t

v t v s

w v w r

INVERSES:

sv = t tt = t uu = t

What about w and r ?

w and r are not inverses.

w w = t and rr = t

t

t

r s t u v w

r t r s

s s t r

t r s t u v w

u u t

v t v s

w v w r t

CANCELLATION PROPERTY:no element is repeated in any row or column

u and w are missing in yellow column

There is a u in blue row

uv must be w

rv must be u w

u

r s t u v w

r t r u s

s s t r

t r s t u v w

u u t w

v t v s

w v w r t

u and v are missing in yellow column

There is a u in blue row

uw must be v

vw must be u v

u

s and u are missingr and w are missing

r and s are missing

r s t u v w

r t r u s

s s t r

t r s t u v w

u u t w v

v t v s u

w v w r t

s r

r w

u s

u is missing

u

r s t u v w

r t r u s

s u s t r

t r s t u v w

u s r u t w v

v w t v r s u

w v u w s r t

Why is the cancellation property useless in completing the remaining four spaces?

v and w are missing from each row and column with blanks.

We can complete the tableusing the associativeproperty.

r s t u v w

r t r u s

s u s t r

t r s t u v w

u s r u t w v

v w t v r s u

w v u w s r t

( r s ) w = r ( s w )

( r s ) w = r ( s w )

( r s ) w = r ( r )

( r s ) w = t

( r s ) w = t w

w

ASSOCIATIVITY

r s t u v w

r t r u s

s u s t r

t r s t u v w

u s r u t w v

v w t v r s u

w v u w s r t

w

v w

v

A GROUP has the following properties:

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