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

Of course, specifying the action in (ii) is completely equivalent to specifying some homomorphism
Φ : H → Aut(N); just set Φ(h) = φh.
6.2.5 Theorem. Given two abstract groups N, H and a homomorphism Φ : H → Aut(N)
define a binary operation on the Cartesian product set N × H, exactly as in (14) and (15).
(17) (n, h) · (n ′ , h ′ ) = (nφh(n ′ ), hh ′ ) for all n, n ′ ∈ N, h, h ′ ∈ H
where φh = Φ(h) ∈ Aut(N). Then G is a group, the external semidirect product, which
we denote by N ×φ H. The inversion operation g ↦→ g −1 in this group has the form (15).
The subsets N = {(n, e) : n ∈ N} and H = {(e, h) : h ∈ H} are subgroups in G that are
isomorphic to N and H. They satisfy the conditions NH = G, N ∩ H = (e), and N is normal
in G. Thus G is the internal semidirect product of these subgroups.
Proof: There is some bother involved in checking that the operation (17) is associative because
the action does not arise via multiplications in some pre-existing group, as it did in 6.2.1. We
leave these routine but tedious calculations to the reader.
The identity element is e = (e, e) = (e N , e H ). The inverse operation is
(n, h) −1 = (φ h −1(n −1 ), h −1 )
In fact, recalling that φe = id N and that φ h −1 = (φh) −1 because Φ : H → Aut(N) is a
homomorphism, (17) gives
(n, h) · (φh−1(n −1 ), h −1 ) = (nφh(φ −1
h (n−1 )), hh −1 ) = (nn −1 , e) = (e, e)
(φh−1(n −1 ), h −1 ) · (n, h) = (φh−1(n −1 )φ −1
h (n), h−1h) = ((φ −1
(n), e) = (e, e)
h (n))−1φ −1
h
In G the maps n ↦→ (n, e) ∈ N and h ↦→ (e, h) ∈ H are isomorphic embeddings of N and H
in G because they are bijections such that
(n, e) · (n ′ , e) = (nn ′ , e) and (e, h) · (e, h ′ ) = (e, hh ′ )
We get G = NH because (n, e) · (e, h) = (nφe(e), eh) = (n, h), and it is obvious that N ∩ H =
{(e, e)}. Normality of N follows because
(n, h)(n ′ , e)(n, h) −1 = (n, e)(e, h)(n ′ , e)(e, h −1 )(n −1 , e)
= (n, e)(φh(n ′ ), h)(e, h −1 )(n −1 , e)
= (n, e)(φh(n ′ ), e)(n −1 , e)
= (nφh(n ′ )n −1 , e)
for all n ′ ∈ N; the lefthand component is back in N so that gNg −1 ⊆ N for any g = (n, h) ∈ G.
When we identify N ∼ = N and H ∼ = H, the action of h ∈ H by conjugation on n ∈ N
matches up with the original action of H × N → N determined by Φ, so the group G we have
constructed is the internal semidirect product N ×φ H of these subgroups. �
The next example is important in geometry.
6.2.6 Example (The Dihedral Groups Dn). These nonabelian groups of order |Dn| = 2n,
defined for n ≥ 2, are the full symmetry groups of regular n-gons. To describe Dn consider a
regular n-gon in the xy-plane, centered at the origin and with one vertex on the positive x-axis,
as shown in Figure 6.2 (where n = 6). Let θ = 2π/n radians, and define the basic symmetry
operations
ρθ = (counterclockwise rotation about the origin by θ radians)
σ = (reflection across the x-axis)
14