Section 14 Factor Groups

1
Section 14 Factor Groups

• Factor Groups from Homomorphisms.
• Theorem
• Let ? G? G be a group homomorphism 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.

2
Examples

• Example
• Consider the map ? Z ? Zn, where ?(m) is the
  remainder when m is divided by n in accordance
  with the division algorithm. We know ? is a
  homomorphism, and Ker (?) n Z.
• By previous theorem, the factor group Z / nZ is
  isomorphic to Zn. The cosets of n Z (nZ, 1n Z,
  ) are the residue classes modulo n.
• Note Here is how to compute in a factor group
• We can multiply (add) two cosets by choosing any
  two representative elements, multiplying (adding)
  them and finding the coset in which the resulting
  product (sum) lies.
• Example in Z/5Z, we can add (25Z)(45Z)15Z
  by adding 2 and 4, finding 6 in 15Z, or adding
  27 and -16, finding 11 in 15Z.

3
Factor Groups from Normal Subgroups

• Theorem
• Let H be a subgroup of a group G. Then left coset
  multiplication is well defined by the equation
• (aH)(bH)(abH)
• If and only if H is a normal subgroup of G.

4
Definition

• Corollary
• Let H be a normal subgroup of G. Then the cosets
  of H form a group G/H under the binary operation
  (aH)(bH)(ab)H.
• Proof. Exercise
• Definition
• The group G/H in the proceeding corollary is the
  factor group (or quotient group) of G by H.

5
Examples

• Example
• Since Z is an abelian group, nZ is a normal
  subgroup. Then we can construct the factor group
  Z/nZ with no reference to a homomorphism. In fact
  Z/ nZ is isomorphic to Zn.

6
Theorem

• Theorem
• Let H be a normal subgroup of G. Then ?? G ? G/H
  given by ?(x)xH is a homomorphism with kernel H.
  
• Proof. Exercise

7
The Fundamental Homomorphism Theorem

• Theorem (The Fundamental Homomorphism Theorem)
• Let ? G ? G be a group homomorphism with kernel
  H. Then ?G is a group, and ? G/H ? ?G given
  by ?(gH) ?(g) is an isomorphism.
• If ?? G ? G/H is the homomorphism given by
  ?(g)gH, then ?(g) ? ?(g) for each g?G.

?
G
?G
?
?
G/H
8
Example

• In summary,
• every homomorphism with domain G gives rise to a
  factor group G/H, and every factor group G/H
  gives rise to a homomorphism mapping G into G/H.
  Homomorphisms and factor groups are closely
  related.
• Example Show that Z4 X Z2 / (0 X Z2) is
  isomorphic to Z4..
• Note that ?1 Z4 X Z2 ? Z4 by ?1(x, y)x is a
  homomoorphism of Z4 X Z2 onto Z4 with kernel 0
  X Z2. By the Fundamental Homomorphism Theorem, Z4
  X Z2 / (0 X Z2) is isomorphic to Z4.

9
Normal Subgroups and Inner Automorphisms

• Theorem
• The following are three equivalent conditions for
  a subgroup H of a group G to be a normal subgroup
  of G.
• ghg-1 ? H for all g ? G and h ? H.
• ghg-1 H for all g ? G.
• gH Hg for all g ? G.
• Note Condition (2) of Theorem is often taken as
  the definition of a normal subgroup H of a group
  G.
• Proof. Exercise.
• Example Show that every subgroup H of an abelian
  group G is normal.
• Note ghhg for all h ? H and all g ? G, so ghg-1
  h ? H for all h ? H and all g ? G.

10
Inner Automorphism

• Definition
• An isomorphism ? G ? G of a group G with itself
  is an automorphism of G. The automorphism ig G ?
  G , where Ig(x)gxg-1 for all x?G, is the inner
  automorphism of G by g. Performing Ig on x is
  callled conjugation of x by g.