Use the Fundamental Theorem of Homomorphism for Groups to prove the following
theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H′ and K′ be normal subgroups of H
and K, respectively. Then
i) H′(H ∩ K′) H′(H ∩ K)
H (H K)
′ ∩ ∩ ′
∩ − ′ ′∩
′ ∩ − ′ ∩ ′
′ ∩ (15)
The situation can be represented by the subgroup diagram below, which explains the
name ‘butterfly’
Let H and K be subgroups of a group G and H′ and K′ be normal subgroups of H
and K, respectively. Then
i) H′(H ∩ K′) H′(H ∩ K)
ii) K′(H′∩ K) K′(H ∩ K)
Fundamental Theorem of Homomorphism for Groups:
Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K (where G/K is a quotient group). If K is a subset of ker(f) then there exists a unique homomorphism h:G/K→H such that f = h φ.
Proof:
since H,K are subgroup of G, (H∩K) is also a subgroup of G. Moreover, (H∩K)⊂K, so (H∩K) is a subgroup of K also.
K' is normal (H∩K) = (H∩K)∩K' is a normal subgroup d (H∩K). Similarly, (H'∩K) is normal in (H∩K).
Consequently product of two normal subgroups, (H∩K') (H'∩K)= L, is a normal subgroup of (H∩K).
Now, (H∩K)CH,H'<H implies that (H∩K) H' is a subgroup of H. Similarly, (H∩K)K' is a subgroup of K.
Define a map: ϕ: (H∩K)H'→ (H∩K)/L by
ϕ(xy) = xL, x∈H∩K,y∈H'
ϕ is well defined, since for x,x1 ∈H∩K,y,y1∈H'
If
xy =x1y1
x1-1x=y1y-1∈(H∩K)∩H'= (H'∩K)
x1-1x∈(H'∩K)⊂L
x1-1x∈L
xL=x1L
ϕ (xy) =ϕ (x1y1)
ϕ is clearly onto.
Now for x,x1 ∈H∩K,y,y'∈H'
ϕ [(xy)(x1y1)]= ϕ [xx1(x1-1yx1)y1]= ϕ [xx1y2y1]
where y2 =x1-1yx1∈H'
=xx1 L
=xLx1 L
=ϕ (xy) ϕ (x1y1)
so ϕ is an epimorphism.
Further,
xy ∈ker ϕ ⟺ϕ (xy) = L
⟺xL =L
⟺x∈L
⟺x=x1y1, with x1 ∈H∩K', y1 ∈H'∩K
thus xy ∈ker ϕ if only if
xy=(x1y1)y
=x1(y1y) ∈(H∩K')H'
Therefore, ker ϕ = (H∩K') H'.
This implies that (H∩K')H'⊲ (H∩K)H'.
Finally, by Fundamental Homomorphism Theorem,
we have (H∩K)H'/(H ∩K')H'≅ (H∩K)/L.
A symmetric argument shows that
(H'∩K)K'⊲ (H∩K)K', and (H∩K)K'/(H'∩K)K'≅(H∩ K)/L.
Consequently, (H∩ K) H'/(H∩K')H'≅ (H∩K) K'/(H'∩K) K'.
Comments