gfaktor from hom
TRANSCRIPT
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Introduction
Let Gb a group and let Hbe asubgroup of G.
If H is the kernel of a group
homomorphisms G Gthen the left cosets of H will be
elements of a group whose binaryoperation is derived from the
group operation of G.
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Introduction (cont.)
+ 0 3 1 4 2 5
0 0 3 1 4 2 5
3 3 0 4 1 5 2
1 1 4 2 5 3 0
4 4 1 5 2 0 3
2 2 5 3 0 4 1
5 5 2 0 3 1 4
K B M
K K B M
B B M K
M M K B
See Tables
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Introduction (Cont.)
Let G G be a grouphomomorphisms with kernelH.
For a G,
is the left coset aHofHand is also the rightcosetHa ofH.
1 |a x G x a
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Introduction (Cont.)
Since these left and right cosets ofHcoincide,we will simply refer to them as cosets ofH.
Now is a group (Property ofHomomorphisms). We associate with each
the coset .
By renaming by the name of theassociate coset, that is, by , we canconsider the cosets to form a group.
G
y G 1
| y x G x a
y G 1 y
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Introduction (Cont.)
This group will be isomorphic to since itis just renamed.
In summary, the cosets of the kernel of agroup homomorphisms form a group
isomorphic to the subgroup of .
The binary operation on the cosets can be
computed in terms of the group operation of .
G
: 'G G
G
'G
G
'G
This group of cosets is the factor group of Gmodulo H, and is denoted by G/H.
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Theorem
Let be a group homomorphisms
with kernel H. Then the cosets of H form agroup, G/H, whose binary operationdefines the product (aH)(bH) of two cosets
by choosing elements a and b from thecosets, and letting
(aH)(bH) = (ab)H.
Also, the map defined by
is an isomorphisms.
: 'G G
: /G H G aH a
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Proof
The first thing we have to worry about isthe definition (aH)(bH) = (ab)H for the
product of two cosets in G/H.
The product is compued by choosing anlement from each of the cosets aHand bH,
and b finding the coset containing theirproduct ab.
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Proof (cont.)
Any time the definition of something involvesmaking choices, we should show that the end
result is independent of the choices made.
We say that a thing is well defined if it isindependent of any choices made in its
computation.
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Proof (cont.)
Thus we start by showing that
(aH)(bH) = (ab)H
gives a well-defined operation on G/H.
Suppose that and
are two other representative elements from
these cosets.
1ah aH
2bh bH
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Proof (cont.)
We must show that
that is, that ab and lie in the same cosetof G/H.
We need only show that
1 2 .ab ah bh
1 2ab H ah bh H
1 2ahbh
Recall that for all 'h e h H
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Proof (cont.)
Thus
.ab
1 2 1 2ah bh a h b h
Thus we do indeed have a well-definedbinary operation on G/H.
' '
a e b e
a b
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Proof (cont.)
We now check the group axioms for G/H.
aH bH cH aH bc H a b c H
Associative
ab cH
ab H cH
.aH bH cH
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Proof (cont.)
aH eH ae H
aH
Indentity
ea H
.eH aH
SoH= eHacts as indentity coset in G/H.
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Proof (cont.)
1 1a H aH a aH eH
Inverses
1aa H
1.aH a H
So is the inverse of aHin G/H.1
a H
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Proof (cont.)
aH bH ab H ab
Homomorphisms
a b
.aH bH
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Proof (cont.)
One to one
a b aH bH Suppose . Then ,
so that aH and bH are the same coset.
Let Then for someand
Onto
.y G
y x
x G
. xH x y
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Example
Considered the map 4:
where is the remainder when m is
devided by 4 in accordance with the division
algoritm.
m
We know that is a homomorphisms.
ker 4 Of courses
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Example (cont.)
Considered the map
4 , 8, 4,0, 4,8,
We know that is aisomorphisms.
4: / 4
1 4 , 7, 3,1,5,9, 2 4 , 6, 2, 2,6,10,
3 4 , 5, 1,3,7,11,
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Example (cont.)
4: / 4
4 0
1 4 1
2 4 2
3 4 3