ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Proof : This is just T91
111. Definition: Quotient group.
Let N be a normal subgroup of the group G. The induced quotient structure
is said to be the quotient G and N and is denoted G/N.
G∼N
9. 5: Group homomorphisms
112. Theorem: The kernel of a group homomorphism is a normal
subgroup .
The kernel of a group homomorphism f : G1 → G2 is a normal subgroup and
the coset are the level sets of f. The equivalence relation induced by f is denoted
∼f and is identical with the one induced by the kernel ∼ ker(f).
Proof : Let N denote the kernel of f. We use one of the criteria for normality
(T 107), namely x −1 Nx ⊆ N. Let h ∈ N and x ∈ G be arbitrary. Then
f(x −1 hx) = f(x −1 )f(h)f(x) = f(x) −1 1Bf(x) = 1B, and so x −1 hx ∈ N.
Let now aN be an arbitrary coset. Then we have that
x ∈ aN ⇔ a −1 x ∈ N ⇔ f(a −1 x) = 1B ⇔ f(a) −1 f(x) = 1B ⇔ f(x) = f(a),
so that aN = {x|f(x) = f(a)}. From this it also follows that the cosets of N
are the same as the equivalence classes for f
113. Definition: The homomorphism induced by a quotient group.
The quotient group induced by the kernel of some homomorphism f is also
called the quotient group induced by f and we can denote it by G/f.
A normal subgroup can thus be constructed from a homomorphism. A little
more surprising is it may be than any normal subgroup may be constructed
this way. But that is what is said in
114. Theorem: Any normal subgroup is the kernel of some homomorphism
(for instance the canonical projection)
Let N be a normal subgroup i G. The canonical projection kN of G on G/N is
a group homomorphism with kernel N.
61