Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

This means there is a natural “parametrization” of points in G by pairs (a, b) in the Cartesian
product space A × B, which at the moment is not equipped with a group structure. The
correspondence A × B ≈ G is effected by the bijection p : A × B → G with p(a, b) = a · b
(product of a and b in G). The labels a and b may be thought of as “coordinates” labeling
all points in G, and it is natural to ask what the group operation looks like when described in
terms of these parameters.
If both A and B are normal subgroups then by 6.1.13 G is their internal direct product and
our question has a simple answer. When we identify G ≈ A × B the group operations take the
familiar form
(a, b) ∗ (a ′ , b ′ ) = (aa ′ , bb ′ ) (a, b) −1 = (a −1 , b −1 )
and the identity element is e = (e, e).
If neither subgroup is normal, the description of the group law in terms of these parameters
can be formidable, and we will not attempt to analyze this situation here. There is, however,
a fruitful middle ground: the case in which just one of the subgroups is normal. This leads to
the notion of semidirect product, which encompasses an enormous set of examples that arise in
geometry and higher algebra.
In this setting we relabel the subgroups as N and H with N the normal subgroup. That
way it will be readily apparent in calculations which group elements lie in the normal subgroup.
Points in G correspond to pairs (n, h) in the Cartesian product set N × H. Furthermore, the
product operation in G can be described explicitly in terms of these pairs. Given elements
g1 = n1h1, g2 = n2h2 in G we must rewrite g1g2 = n1h1 · n2h2 as a product n ′′ h ′′ with
n ′′ ∈ N, h ′′ ∈ H. The problem is, essentially, to move h1 to the other side of n2, keeping track
of the damage if they fail to commute. This is easy. Because N ⊳ G we may multiply and
divide by h1 to get
n1h1n2h2 = n1(h1xh −1
1 ) · h1h2 = n ′′ · h ′′
in which n ′′ ∈ N because h1Nh −1
1 ⊆ N. Identifying each gi with its corresponding coordinate
pair (ni, hi) in N × H, the group multiplication law takes the form
(13)
(n1, h1) · (n2, h2) = (n1(h1n2h −1
1 ), h1h2)
= (n1φh1(n2), h1h2)
where φh : N → N is the conjugation operation φh(n) = hnh −1 for any h ∈ H. The normal
subgroup N is invariant under the action of any conjugation αg(n) = gng −1 , g ∈ G, and φh
is just the restriction φh = αh|N of the inner automorphism αh to the set N. Obviously
φh ∈ Aut(N) for each h ∈ H, and we obtain an action H × N → N of H on N such that
φe = id N φh1h2 = φh1 ◦ φh2 φ h −1 = (φh) −1 (inverse of the operator φh)
which means that the correspondence Φ : H → Aut(N) given by Φ(h) = φh is a group homomorphism.
Conversely, as we noted in our previous discussion of group actions (Chapter 4),
every such homomorphism Φ determines an action of H on N by automorphisms. There is a
one-to-one correspondence between actions H ×N → N and homomorphisms Φ : H → Aut(N).
It is evident from (13) that the original group G can be reconstructed from its subgroups
N, H if we know the action of H on N via conjugation operators φh. In this way G has been
realized as an “internal semidirect product” of N by H – i.e. as the Cartesian product set N ×H
equipped with the multiplication law (13) determined by φh. We summarize these remarks as
follows.
6.2.1 Proposition (Internal Semidirect Product). Let G be a group and N, H two subgroups
such that
(i) N is normal in G (ii) NH = G (iii) N ∩ H = (e)
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