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PROPOSITION ON SUBGROUPS
Section 3.8
3.8.1Proposition: Let beasubgroupofthe
group ,andlet , ∈ .Thenthefollowing
conditionsareequivalent:
(1)
Cosets, Normal Subgroups, and Factor Groups
(2)
(3)
(4)
A COROLLARY
3.8.2Corollary: Let beasubgroupofthe
group .Therelation~ definedon bysetting
~ if
,forall , ∈ ,isan
equivalencerelationon .
;
⊆
;
∈
;
∈
.
COSETS
3.8.3Definition: Let beasubgroupofthegroup
,andlet ∈ .Theset
∈ |
forsome ∈
iscalledtheleftcoset of in determinedby .
Similarly,therightcoset of in determinedby
istheset
∈ |
forsome ∈
.
Thenumberofleftcosets of in iscalledthe
index of in ,andisdenotedby : .
COSET MULTIPLICATION AND THE GROUP OF LEFT COSETS
3.8.4Proposition: Let beanormal
subgroupof ,andlet , , , ∈ .If
and
,then
.
FACTOR GROUP
3.8.6Definition: If isanormalsubgroupof
,thenthegroupofleftcosets of iscalled
thefactorgroup of determinedby .Itwill
bedenotedby / .
3.8.5Theorem: If isanormalsubgroupof
,thenthesetofleftcosets of formsagroup
underthecoset multiplicationgivenby
forall ,
∈ .
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FACTOR GROUPS AND HOMOMORPHISMS
3.8.7Proposition: Let beanormalsubgroupof .
(a) Thenaturalprojection ∶ → / definedby
,forall ∈ ,isagrouphomomorphism,
.
andker
(b) Thereisaone‐to‐onecorrespondencebetweenthe
subgroupsof / andsubgroups of with ⊇ .
Specifically,if isasubgroupof / ,then
isthe
correspondingsubgroupof ;if isasubgroupof with
⊇ ,then
isthecorrespondingsubgroupof / .
Underthiscorrespondence,normalsubgroups
correspondtonormalsubgroups.
THE FUNDAMENTAL HOMOMORPHISM THEOREM
3.8.9Theorem(Fundamental
HomomorphismTheorem): Let , be
groups.If ∶
→
isahomomorphism
with
ker
,then ⁄ ≅
.
EQUIVALENT CONDITIONS FOR A NORMAL SUBGROUP
3.8.8Proposition: Let besubgroupof .
Thefollowingconditionsareequivalent.
(1)
isanormalsubgroupof ;
(2)
forall ∈ ;
(3) forall ,
product
∈ ,
;
(4) forall ,
∈ .
∈ ,
isthesettheoretic
∈
ifandonlyif
SIMPLE GROUPS
3.8.10Definition: Thenontivial group is
calledasimple groupifithasnoproper
nontrivialsubgroups.
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