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Week 8
Factor groups (From Homomorphisms.)
Let  : G  G be a homo. and H  Ker  . Then  a  G, aH  Ha .
Let G H be the set of left cosets of H in G and define aH bH   abH . It is
well-defined. Hb  bH !
Note: aH    a and bH    b .
G
H
is read as “G over H”, “G modulo H” or “G mod H”.
(*) G H is a group.
Theorem
Let  : G  G be a group homo. with kernel H . Then the cosets of
H form a factor group, G
H
, where aH bH   abH . Also, the map  : G H   G
defined by  aH    a is an isomorphism. Both coset multiplication and  are
well-defined, independent of the choices a and b from the cosets.
Proof.
 aH bH    abH    ab   a b   aH  bH .
  a,  aH , s.t.  aH    a  . (Onto)
 aH    e   a   e  a  H  aH  H .
e.g.  : Z  Z n ,  m  r if m  qn  r, 0  r  n .
Ker   nZ
Cosets: nZ , 1  nZ , …, n 1  nZ . (The residue classes modulo n)
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The group of cosets Z nZ .
 : Z nZ  Z n (Isomorphism).
Note: We also use Z n to denote the factor group Z nZ . G H is read as factor group
of G modulo H
Theorem. (From Normal subgroups)
Let H  G . Then left coset multiplication is well-defined by aH bH   abH
if and only if H  G .
Proof.
(  )  a  H , consider aH and let x  aH .
Since a 1  a 1 H  xa1  aH a 1 H   H  x  Ha .
Similarly,  y  Ha , y  h1a , y 1  a 1h1 1 , y 1  a 1 H
 y 1a  a 1 H aH   H
 a 1 y  H  y  aH . Hence H  G .
(  )  x  aH and y  bH , we claim： xH  yH   abH .
 z  xH  yH  , z  xh1  yh2   ah3 h1bh4 h2  ah1' bh2'  abh3'  abH .
 z  abH , z  abh  ah1h11bh  ah1bh2' h  aH bH  .
Corollary
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Let H  G . Then the cosets of H in G from a group G H under the
operation aH bH   abH .
Proof.
It is easy to check.
Defn.
G
H
is the factor group (or quotient group) of G by H.
e.g.
nZ  Z  Z nZ is a group.
e.g. Let c  5.8
c  x | x   and x  0, c  c    .
  c ,  is a group.
c  5.8 7.5  c  1.2  6.3c  0.5 .
 c  

 3c,  2c,  c, 0, c, 2c, 3c 

c
 c
cZ 
 x  0, c, x  c  is a coset of  c  in  .
given by  x  xH is a homo with Ker   H .
Theorem
H G  :G G
Theorem
(The fundamental homo. thm.)
H
Let  : G  G  be a group homo. with Ker ( )  H . Then  : G H   G given
by gH   g   g  G is an isomorphism. If  : G  G H is a homo. given by
 g   gH , then  g      g   g  G or      .
Proof.
K e r   G .  G H is a group.
 is a homo.   is a homo.
Ker   H .   is 1-1. The others are easy to check.
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 (1) ghg 1  H  g  G and h  H .

H  G   (2) gHg 1  H  g  G.
 (3) gH  Hg  g  G.

(＊) In order to check H  G , it suffices to show that  g  G gHg 1  H . (Why?)


Since  h  H , h  g g 1hg g 1  g g 1hg 1  g 1  gh1 g 1  gHg 1 .
1
H  gHg 1 whenever gHg 1  H .
Note：  g  G , gHg 1 is also a subgroup of G and | H |  | gHg 1 | .
(＊＊) A normal subgroup plays an important role in finding a factor group.
Theorem (Intersection of normal subgroups)
Let H 1 and H 2 be two normal subgroups of G . Then H1  H 2  G .
Proof.
 g  G , g H1  H 2 g 1  H1 and g H 1  H 2 g 1  H 2 .
Hence g H1  H 2 g 1  H1  H 2 .
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Theorem Let S be a subset of a group G then there exists a normal subgroup of
smallest order which contains S.
Proof.
Let K   H i , H i  S and H i  G .
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iI
Defn. (Automorphism)
An isomorphism  : G  G of a group G with itself is an automorphism of G .
The automorphism i g x  gxg 1  x  G is the inner automorphism of G by “ g ”.
gxg 1 is known as the conjugate of x .
x  gxg 1
 conjugation
Defn. (Conjugate subgroup)
A subgroup K of G is a conjugate subgroup of H if K  i g H  for some
g  G , i.e. K  gHg 1 .
Defn. (Commutator subgroup)
Let G be a group. An element of G that can be expressed in the form aba 1b 1
for some a, b  G is a commutator of G . The commutator subgroup C of G is the
smallest normal subgroup which contains all commutators of G .
Ex. G C is abelian.
Theorem
Proof.
C is normal
in G .
It suffices to show that for every g , a , b in G , g aba 1b 1 g 1 is also in


1
1 1
1 1
C .(?) By g aba b  ga bga  gb g = ga bga  b 1bgb 1 g 1
1
= ga bga 1 b 1 bgb 1 g 1  C , we conclude the proof.
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Ex. Solution



aba 1b 1C  C  aC bC  a 1C b 1C  C  aC bC   bC aC  .
Remark
Defn.
A subgroup H of G is said to be a characteristic subgroup of G if for
all automorphisms  of G ,  H   H .
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Theorem
Proof.
If H is a characteristic subgroup of G , then H  G .
Since the mapping defined by i g x  gxg 1 is an automorphiam and
i g x   gxg 1  H for each x  H . Hence gHg 1  H for each g  G . This implies
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that H  G .
Theorem
of G .
Proof.
The commutator subgroup C of G is also a characteristic subgroup
This is a direct conclusion of
 aba 1b 1    a  b a 1  b 1    a  b  a 1  b 1 . (  C  C .)
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