John Stillwell - Naive Lie Theory.pdf - Index of

30 2 Groups
and
h ∈ ker ϕ ⇒ ϕ(h)=1
⇒ ϕ(h) −1 = 1
⇒ ϕ(h −1 )=1
⇒ h −1 ∈ ker ϕ.
2. ker ϕ is a normal subgroup of G, because, for any g ∈ G,
h ∈ ker ϕ ⇒ ϕ(ghg −1 )=ϕ(g)ϕ(h)ϕ(g −1 )=ϕ(g)1ϕ(g) −1 = 1
⇒ ghg −1 ∈ ker ϕ.
Hence g(ker ϕ)g −1 = ker ϕ,thatis,kerϕ is normal.
3. Each g ′ = ϕ(g) ∈ G ′ corresponds to the coset g(ker ϕ).
In fact, g(ker ϕ)=ϕ −1 (g ′ ), because
k ∈ ϕ −1 (g ′ ) ⇔ ϕ(k)=g ′ (definition of ϕ −1 )
⇔ ϕ(k)=ϕ(g)
⇔ ϕ(g) −1 ϕ(k)=1
⇔ ϕ(g −1 k)=1
⇔ g −1 k ∈ ker ϕ
⇔ k ∈ g(ker ϕ).
4. Products of elements of g ′ 1 ,g′ 2 ∈ G′ correspond to products of the
corresponding cosets:
g ′ 1 =ϕ(g 1),g ′ 2 =ϕ(g 2) ⇒ ϕ −1 (g ′ 1 )=g 1(ker ϕ),ϕ −1 (g ′ 2 )=g 2(ker ϕ)
by step 3. But also
g ′ 1 = ϕ(g 1),g ′ 2 = ϕ(g 2) ⇒ g ′ 1 g′ 2 = ϕ(g 1)ϕ(g 2 )=ϕ(g 1 g 2 )
⇒ ϕ −1 (g ′ 1 g′ 2 )=g 1g 2 (ker ϕ),
also by step 3. Thus the product g ′ 1 g′ 2 corresponds to g 1g 2 (ker ϕ),
which is the product of the cosets corresponding to g ′ 1 and g′ 2 respectively.
To sum up: a group homomorphism ϕ of G onto G ′ gives a 1-to-1 correspondence
between G ′ and G/(ker ϕ) that preserves products, that is, G ′
is isomorphic to G/(ker ϕ).
This result is called the fundamental homomorphism theorem for
groups.