Mathematical foundation of computer science

2. GROUP HOMOMORPHISM
Group homomorphism g[>] are
functions that preserve group structure. A
function a:G->H between two groups
(G,.) and (H,*) is called a homomorphism
if the equation,
a(g . k)=a(g) * a(k)
hold for all element g , k in G.

3. Two groups G and H are called isomorphic if
there exist group homomorphism a:G-> H and
b:H->G , such that applying the two functions
one after in each of the two possible orders
gives the identity functions of G and H. That is
, a(b(h))=h and b(a(g))=g for any g in G and h
in H.

4. Example:
Proving that g . g=1g for some
element g of G is equivalent to
proving that a(g) *a(g)=1H, applying
a to 1st equation yields the 2nd and
applying b to 2nd gives back the
1stequation.

5. SUBGROUPS
Informally, a subgroup is a group
H contained within a bigger one, G.[30]
concretely, the identity element of G is
contained in H, and whenever h1 are in H,
then so are h1. h2 and h1-1,so the elements
of h , equipped with the group operation
on G restricted to H , indeed form a group.

6. Example:
 The identity and the rotations constitute
a subgroup R={id,r1,r2,r3}
 Complementary rotations are 270º for
90º, 180º for 180º and 90º for 270º.

7. ROTATION

8. 90º ROTATION

9. 180º ROTATION

10. 270º ROTATION

11.  Theorem:
f:G→His a group homomorphism.
⇒The kernal of f , Ker f:= {g ∈ G | f(g) =e}is a
subgroup of G
Proof:
Note: f(e) f (ee) f(e)f(e) Hence f(e)is the
identity in H That is Kerf/φ
Als of(b)f(b−1) =f(bb−1) =f(e)
By uniqueness off(b)−1.
we find f(b−1) =f(b)−1 for any a , b ∈ Kerf.
we have f(a−1 =f(a)f(b−1)s the identity
inHThusa−1erfHence Kerfs a subgroup of G .