A GROUP has the following properties:

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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, xy is in G.
The IDENTITY is i for all x in G, ix xi x
Every element in G has an INVERSE km i pp
i rr i ss i
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G has ASSOCIATIVITY for every x, y, and z in
G, (xy)z x(yz) for example
( kp ) 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 have COMMUTATIVITY pr rp
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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
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
H i , k , m
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
H i , k , m
r
s
p
r , s , p
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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H i, k, m a subgroup
The COSETS of H are
iH ii, ik, im i,k,m
kH ki, kk, km k,m,i
mH mi,mk, mmm,i,k
pH pi, pk, pm p,r,s
rH ri, rk, rm r,s,p
sH si, sk, sm s,p,r
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The cosets of a subgroup form a group
A B A A B B B A
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M A,B,C,D,E,F,G,H is a noncommutative
group. N B, C, E, G is a subgroup of M
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The cosets of N B, C, E, G are
AN D,F,A,H
BN C,G,B,E
CN G,E,C,B
DN F,H,D,A
EN B,C,E,G
FN H,A,F,D
GN E,B,G,C
HN A,D,H,F
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Rearrange the elements of the table so that
members or each coset are adjacent and see the
pattern!
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Q is a commutative group R c, f, I is a
subgroup of Q
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The cosets of R d,g,a e,h,b, c,f,I
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The cosets of a subgroup partition the group
ie every member of the group belongs to
exactly one coset.
a b c d e
f g h i
LAGRANGES THEOREM the order of a subgroup is a
factor of the order of the group.
(The order of a group is the number of
elements in the group.)
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If we rearrange the members of Q, we can see that
the cosets form a group
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familiar examples of groups
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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,
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)
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11,
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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,
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)
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11,
and coset (add 2 to every member of T)
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R2
example 2 The set of all points on the plane
with operation defined
The identity is the origin.
L
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Theorem Every group has the cancellation
property.
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Theorem Every group has the cancellation
property.
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 as
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Theorem Every group has the cancellation
property.
In a group, every element has an inverse and
you have associativity.
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EXERCIZE
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COMPLETE THE TABLE TO MAKE A GROUP
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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
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The IDENTITY is t
tr r
r
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The IDENTITY is t
tr r
ts s
s
r
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The IDENTITY is t
tr r
ts s
s
r
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The IDENTITY is t
and
s
r
u
v
w
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sv t
s and v are INVERSES
vs t
t
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u is its own inverse
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INVERSES sv t tt t uu t
What about w and r ?
t
w and r are not inverses.
w w t and rr t
t
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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
u
uv must be w
rv must be u
w
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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
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u is missing
u
r
s
w
r
u
s
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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 table using the
associative property.
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ASSOCIATIVITY
( r s ) w r ( s w )
w
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w
v
w
v