1. A mapping that preserves algebraic structure
HOMOMORPHISM
A mapping that preserves algebraic structure

2. range
domain x
T(x) y
T(y)
x * y
T(x) # T(y)
Operation: *
Operation: #

3. homomorphism range domain T( x*y ) = T(x) # T(y) x T(x) y T(y) x * y
Operation: *
Operation: #
T( x*y ) = T(x) # T(y)

4. DOMAIN a d b e c f
RANGE P Q R
T(e*c)=
T(e) #T(c)

5. If we rearrange the order of the elements of the table on the left
we can clearly see the effect of the homomorphism - the
preservation of the structure of the domain.

6. DOMAIN a d b e c f
RANGE P Q R

7. kernel
DOMAIN a d b e c f
identity in range
RANGE P Q R

8. DOMAIN a d b e c f
RANGE
P = identity Q R
a d
= kernel
a+b= d+b=
a+c= d+c=
b e
c f
A coset of the kernel
A coset of the kernel