Transcript Abstract Algebra

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.
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, 1+n 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 (2+5Z)+(4+5Z)=1+5Z by adding 2 and 4,
finding 6 in 1+5Z, or adding 27 and -16, finding 11 in 1+5Z.
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.
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.
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.
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
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
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.
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.
1. ghg-1  H for all g  G and h  H.
2. ghg-1 = H for all g  G.
3. 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: gh=hg for all h  H and all g  G, so ghg-1 = h  H for all h  H
and all g  G.
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.