Now we consider two multiplicative groups  G,   and  H ,  with identity
elements eG  G and eH  H respectively.
Def: A homomorphism from G to H is a mapping  : G  H such that
  a  b     a    b 
for all
a, b  G
Def: An isomorphism from G to H is a homomorphism from G to H that
is both one-to-one and onto.
Example:  G,     ,  
and
 H ,      ,  define  :    
by   n   2 n
Def: Let
 :G  H
be a homomorphism. Then:
1. The image of  is
2. The kernel of  is
Im    h  H | h    g  for some g  G
Ker     g  G |   g   eH 
Example (from above): Find the image and kernel of  .
Proposition: Let
1.
 :G  H
be a homomorphism. Then
  eG   eH
2. For all g  G,   g 1     g 
3. Im   is a subgroup of H
4. Ker   is a subgroup of G
5.  is one-to-one if and only if Ker    eG 
6. For all k  Ker   and for all x  G, x  k  x 1  Ker  
1
Abstract Algebra (continued)
Definition: A normal subgroup of a group  G,   is a subgroup N of G
such that
x  n  x 1  N for all x  G and all n  N .
Note: How does this apply if the group G is abelian?
What about the trivial subgroups? Are they normal?
Definition: Let K be a subgroup of a group  G,   and let x  G . The left
coset of K in G containing x is the set
x  K  x  k | k  K
Proposition: Let K be a subgroup of the group  G,   . Then the set
P   x  K | x  G
of all left cosets of K in G forms a partition of G.
Lagrange’s Theorem: Let K be a subgroup of the finite group  G,   and
let P denote the set of all left cosets of K in G. Then
G  KP
Def: Let  G,   be a group and let N be a normal subgroup of G.
Consider the set of all left cosets of N in G and denote it by G/N:
G / N   x  N | x  G
Theorem: Let  G,   be a group and let N be a normal subgroup of G.
Define the operation, also denoted by  on the set G/N by
 x  N   y  N    x  y  N
for all x, y  G
Then:
1. This product of cosets is well defined.
2.  G / N ,   is a group with identity eG / N  eG  N  N
1
1
3. For each x  G,  x  N   x  N
4. The mapping v : G  G / N defined by v  x   x  N , for all x  G is a
surjective homomorphism from G to G/N, and Ker  v   N
The group  G / N ,   is called the quotient group (or factor group) of G
modulo N, and the surjective homomorphism v is said to be the natural
homomorphism from G to G/N (its quotient group).
The Isomorphism Theorem for Groups
A homomorphic image of the group  G,   is any group  G,  with the
property that there exists a homomorphism  : G  G from G onto G
Ex: For each integer m,  m is a homomorphic image of  ,   .
Theorem: Let  G,   and  H ,  be groups and let  : G  H be a
homomorphism. Then Ker   is a normal subgroup of G and Im   is a
subgroup of H which is isomorphic to the quotient group G / Ker  
Rings and Fields
Definition: A ring is a triple  R, ,  where R is a set and “+” and “.” are
two binary operations such that:
1) (R,+) is an Abelian group
2)
3)
a, b, c  R, a  (b  c)  ( a  b)  c
a, b, c  R, a  (b  c )  a  b  a  c and (a  b)  c  a  c  b  c .
If there is a multiplicative identity, we say that “R is a ring with unity.”.

is a ring with unity, but E (even integers) is a ring without unity.
Notation: U(R)= the set of all units in R. (“unit” is an element that has an
inverse!)
Definition: Ring homomorphism
If R and R’ are rings,  : R  R ' is a ring homomorphism if
 ( x  y )   ( x)   ( y )
 ( x  y )   ( x)   ( y )
Definition: An Ideal I in R is a subring of R which is closed under
external-internal multiplication. x  R, i  I ;
xi  I , ix  I .
Principal ideal generated by a in a commutative ring with unity R:
 a   ar : r  R
Let R be a ring and
a0
be an element in R. If there exists a nonzero
element b in R such that ab=0, then a is a “zero divisor”.
Quotient Rings: Similar to Quotient Groups.
Definition: If there are positive integers n such that nx=0 for all x in R,
then the smallest such n is called “the characteristic of R”. If no such
positive integer exists, we say “R has characteristic 0”.
Example: Real numbers and Z have characteristic 0.
For
6 ,
n=6 is the smallest integer that works; it has characteristic 6.
Definition: Integral Domain
Let D be a ring. D is an integral domain if
1) D is a commutative ring
2) D has unity e and e is nonzero.
3) D has no zero divisors.
Example: Z is an integral domain, but E is not (no unity). The set of
rational numbers and real numbers are integral domains. M n    or
are not integral domains (for
n  2 ).
Mn  R
Definition: Let F be a ring. Then F is a field if:
1) F is a commutative ring.
2) F has a nonzero unity e.
3) Every nonzero element in F has a multiplicative inverse.
Example: Rational numbers, Real numbers, Complex numbers are fields.
Z is not.
Every field is an integral domain.
Fact:
n
is a field if and only if n is prime.