1 The Direct Product of Groups
Given groups G and H, the external direct productG×H is the set of ordered pairs {(g,h)∣g∈G,h∈H} equipped with the componentwise operation (g1,h1)(g2,h2)=(g1g2, h1h2).
G×H is a group. The identity is (eG,eH), and the inverse of (g,h) is (g−1,h−1).
Associativity follows from that of G and H: (g1,h1)((g2,h2)(g3,h3))=(g1,h1)(g2g3,h2h3)=(g1g2g3,h1h2h3), which equals ((g1,h1)(g2,h2))(g3,h3). The identity and inverse properties are verified componentwise. □
In G×H, define Gˉ={(g,eH)∣g∈G} and Hˉ={(eG,h)∣h∈H}. Then:
Gˉ⊴G×H and Hˉ⊴G×H.
Gˉ∩Hˉ={(eG,eH)}.
GˉHˉ=G×H.
Gˉ≅G and Hˉ≅H.
For (1), take any (g′,h′)∈G×H and (g,eH)∈Gˉ. Then (g′,h′)(g,eH)(g′,h′)−1=(g′gg′−1,eH)∈Gˉ, so Gˉ⊴G×H. The argument for Hˉ is identical. Parts (2), (3), and (4) follow directly from the definitions. □
2 The Internal Direct Product
Let G be a group with normal subgroups N and K. We say G is the internal direct product of N and K, written G=N×K, if the following three conditions hold:
N⊴G and K⊴G.
N∩K={e}.
NK=G.
If N and K are normal subgroups of G with N∩K={e} and NK=G, then the map φ:N×K→G defined by (n,k)↦nk is a group isomorphism.
We first show that every element of N commutes with every element of K. For n∈N and k∈K, consider the commutator nkn−1k−1. Since K⊴G, we have nkn−1∈K, so nkn−1k−1=(nkn−1)k−1∈K. Since N⊴G, we have kn−1k−1∈N, so nkn−1k−1=n(kn−1k−1)∈N. Thus nkn−1k−1∈N∩K={e}, giving nk=kn.It follows that φ is a homomorphism: φ((n1,k1)(n2,k2))=φ(n1n2,k1k2)=n1n2k1k2=n1k1n2k2=φ(n1,k1)φ(n2,k2).
Surjectivity follows from NK=G. For injectivity, if nk=e then n=k−1∈N∩K={e}, so n=k=e. □
If G=N×K is an internal direct product, then every element of N commutes with every element of K: nk=kn for all n∈N, k∈K.
3 Generalized Direct Products
A group G is the internal direct product of normal subgroups N1,…,Nr, written G=N1×⋯×Nr, if:
Each Ni⊴G.
Ni∩(N1⋯Ni−1Ni+1⋯Nr)={e} for i=1,…,r.
N1N2⋯Nr=G.
We have Z/6Z≅Z/2Z×Z/3Z. Indeed, N={0,3}≅Z/2Z and K={0,2,4}≅Z/3Z are normal subgroups of Z/6Z with N∩K={0} and N+K=Z/6Z.
If gcd(m,n)=1, then Z/mnZ≅Z/mZ×Z/nZ (the group-theoretic Chinese Remainder Theorem).
4 Motivation for the Semidirect Product
In a direct product G=N×K, the elements of N and K commute with each other. However, many groups cannot be decomposed in this way.
Consider the dihedral group Dn (the symmetry group of a regular n-gon, of order 2n). The rotation subgroup R=⟨r⟩≅Z/nZ is normal, and the subgroup S={e,s}≅Z/2Z generated by a reflection satisfies R∩S={e} and RS=Dn. But S is generally not normal, and rs=sr (in fact srs−1=r−1).
This situation motivates the semidirect product, which generalizes the direct product by allowing one factor to act non-trivially on the other.
5 Definition of the Semidirect Product
Let N and H be groups and φ:H→Aut(N) a homomorphism. The semidirect product of N and H with respect to φ, denoted N⋊φH, is the set N×H equipped with the operation (n1,h1)(n2,h2)=(n1⋅φ(h1)(n2), h1h2).
Associativity: Using the facts that each φ(h) is an automorphism and φ is a homomorphism: ((n1,h1)(n2,h2))(n3,h3)=(n1φ(h1)(n2), h1h2)(n3,h3)=(n1φ(h1)(n2)⋅φ(h1h2)(n3), h1h2h3)=(n1φ(h1)(n2)⋅φ(h1)(φ(h2)(n3)), h1h2h3)=(n1φ(h1)(n2φ(h2)(n3)), h1h2h3)=(n1,h1)(n2φ(h2)(n3), h2h3)=(n1,h1)((n2,h2)(n3,h3)).
Identity: (eN,eH) serves as the identity, since (eN,eH)(n,h)=(eN⋅φ(eH)(n),h)=(n,h).Inverse: (n,h)−1=(φ(h−1)(n−1),h−1), as one verifies by direct computation. □
In G=N⋊φH:
Nˉ={(n,eH)∣n∈N}⊴G and Nˉ≅N.
Hˉ={(eN,h)∣h∈H}≤G and Hˉ≅H (but Hˉ need not be normal).
Nˉ∩Hˉ={(eN,eH)} and NˉHˉ=G.
(eN,h)(n,eH)(eN,h)−1=(φ(h)(n),eH). That is, Hˉ acts on Nˉ via φ.
6 Recognition of Internal Semidirect Products
Suppose a group G satisfies:
N⊴G and H≤G.
N∩H={e}.
NH=G.
Define φ:H→Aut(N) by φ(h)(n)=hnh−1. Then G≅N⋊φH.
Since NH=G and N∩H={e}, every element g∈G can be written uniquely as g=nh with n∈N and h∈H. Define Φ:N⋊φH→G by (n,h)↦nh.This is a homomorphism: Φ((n1,h1)(n2,h2))=Φ(n1h1n2h1−1,h1h2)=n1h1n2h1−1⋅h1h2=n1h1n2h2=Φ(n1,h1)Φ(n2,h2).
Surjectivity follows from NH=G, and injectivity from N∩H={e}. □
7 Examples of Semidirect Products
Dn≅Z/nZ⋊φZ/2Z, where φ(1)(k)=−k (negation in Z/nZ).
S3≅A3⋊⟨(12)⟩≅Z/3Z⋊Z/2Z≅D3.
Let p<q be primes with q≡1(modp). Then Aut(Z/qZ)≅(Z/qZ)× has order q−1, which is divisible by p, so it contains an element α of order p. Setting φ:Z/pZ→Aut(Z/qZ) with φ(1)=α, the semidirect product Z/qZ⋊φZ/pZ is a non-abelian group of order pq.
When φ is the trivial homomorphism (φ(h)=idN for all h∈H), the semidirect product reduces to the direct product: N⋊φH=N×H.
8 Summary
Direct productN×K: both N⊴G and K⊴G, with N∩K={e} and NK=G. Elements of N and K commute.
Semidirect productN⋊H: N⊴G and H≤G (not necessarily normal), with N∩H={e} and NH=G. The subgroup H acts on N by conjugation.
The semidirect product depends on the choice of homomorphism φ:H→Aut(N). Different choices of φ generally yield non-isomorphic groups.
The direct and semidirect products are the fundamental methods for assembling groups from known building blocks — or, conversely, for decomposing a group into simpler pieces. They are indispensable tools for understanding group structure.