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A GROUP has the following properties:
• Closure
• Associativity
• Identity
• every element has an Inverse
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mkisrpp
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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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srpmki*
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srpmki*
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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srpmkii
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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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srpmki*
H = { i, k, m }is a SUBGROUP
imkrpss
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mkisrpp
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srpmkii
srpmki*
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 }
H = { i, k, m }is a SUBGROUP
imkrpss
kimpsrr
mkisrpp
psrkimm
rpsimkk
srpmkii
srpmki*
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 }
Example: to form the coset r H
H = { i, k, m }is a SUBGROUP
imkrpss
kimpsrr
mkisrpp
psrkimm
rpsimkk
srpmkii
srpmki*
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 }
Example: to form the coset r H
H = { i , k , m }r r r r
H = { i, k, m }is a SUBGROUP
imkrpss
kimpsrr
mkisrpp
psrkimm
rpsimkk
srpmkii
srpmki*
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 }
Example: to form the coset r H
H = { i , k , m }r r r r
= { r , s , p } s p r
imkrpss
kimpsrr
mkisrpp
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rpsimkk
srpmkii
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.
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”
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.
If
then
yx
yaaxaa
ayaaxa
ayax
)()(
)()(11
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
