Direct and Semidirect Products

Two ways of assembling a group from smaller pieces — and, read backwards, two ways of recognising that a given group decomposes. The direct product glues factors that ignore each other; the semidirect product lets one factor act on the other. This page builds on subgroups-cosets.md (normal subgroups) and homomorphisms.md (automorphisms), and its Lie-algebra analogue reappears in lie-algebras.md.

Direct product

The (external) direct product $G_1\times G_2$ is the set of ordered pairs $(g_1,g_2)$ with componentwise multiplication: $$(g_1,g_2)\cdot (g_1^{\prime },g_2^{\prime })=(g_1{g}_1^{\prime },g_2{g}_2^{\prime }).$$ Both factors embed as normal subgroups $G_1\times \{e\}$ and $\{e\}\times G_2$, they intersect trivially, together generate the product, and they commute elementwise. Orders multiply: $|G_1\times G_2|=|G_1||G_2|$. The Lie-algebra analogue is the direct sum ${\mathfrak{g}}_1\oplus {\mathfrak{g}}_2$ with $[{\mathfrak{g}}_1,{\mathfrak{g}}_2]=0$.

Internal recognition criterion. If $G$ has normal subgroups $N,M>◃G$ with $N\cap M=\{e\}$ and $NM=G$, then $G\cong N\times M$ via $(n,m)\mapsto nm$.

Example: ${\mathbb{Z}}_6\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_3$ (since $\gcd (2,3)=1$); more generally ${\mathbb{Z}}_{mn}\cong {\mathbb{Z}}_m\times {\mathbb{Z}}_n$ iff $\gcd (m,n)=1$ — the Chinese remainder theorem in group form, foundational to cyclic-abelian.md.

Semidirect product

The semidirect product $N{⋊}_{\varphi }H$ combines a normal factor $N$ with a factor $H$ that acts on $N$ by automorphisms. The data is a homomorphism $$\varphi :H\to \mathrm{Aut}(N),h\mapsto {\varphi }_h.$$ The underlying set is pairs $(n,h)$, but the multiplication is twisted by $\varphi$: $$(n_1,h_1)\cdot (n_2,h_2)=(n_1{\varphi }_{h_1}(n_2),h_1{h}_2).$$ The second entry $n_2$ is transformed by ${\varphi }_{h_1}$ before combining. The inverse is $(n,h{)}^{-1}=({\varphi }_{h^{-1}}(n^{-1}),h^{-1})$.

Structural properties.

• $N\cong \{(n,e)\}$ is normal (the kernel of the projection $(n,h)\mapsto h$).
• $H\cong \{(e,h)\}$ is a subgroup, generally not normal.
• Inside the product, conjugation realises the action: $hn{h}^{-1}={\varphi }_h(n)$.
• When $\varphi$ is trivial (${\varphi }_h={\mathrm{id}}_N$ for all $h$), the twist vanishes and $N⋊H=N\times H$ — the direct product is the special case.

Internal recognition (splitting) criterion. If $G$ has a normal subgroup $N$ and a subgroup $H$ with $N\cap H=\{e\}$ and $NH=G$ — a split extension — then $G\cong N⋊H$ with ${\varphi }_h(n)=hn{h}^{-1}$ (conjugation inside $G$).

The extension problem

Both products answer a special case of the general extension problem: given $N$ and $Q$, classify all $G$ with a normal subgroup $\cong N$ and quotient $\cong Q$, $$1\to N\to G\to Q\to 1.$$

• The extension splits (is a semidirect product) iff the quotient map has a section — a subgroup of $G$ mapping isomorphically onto $Q$.
• Not every extension splits. ${\mathbb{Z}}_4$ is a non-split extension of ${\mathbb{Z}}_2$ by ${\mathbb{Z}}_2$: it has the normal subgroup $\{0,2\}\cong {\mathbb{Z}}_2$ with quotient ${\mathbb{Z}}_2$, but no complementary ${\mathbb{Z}}_2$ (its only element of order $2$ is $2$, already inside $N$). The split extension with the same data is $V_4={\mathbb{Z}}_2\times {\mathbb{Z}}_2$ instead. So ${\mathbb{Z}}_4\ncong {\mathbb{Z}}_2⋊{\mathbb{Z}}_2$ for any action.
• Central (abelian-$N$) extensions are classified by the cohomology group $H^2(Q;N)$; the split ones are the zero class. This cohomological viewpoint reappears physically as the origin of projective representations (see topology-covers.md) and anomalies (see physics-bridge.md).

Examples

• Dihedral group $D_n={\mathbb{Z}}_n⋊{\mathbb{Z}}_2$: the reflection acts on the rotation by inversion, $\varphi (r)=r^{-1}$. Symmetries of a regular $n$-gon.
• Symmetric group $S_n=A_n⋊{\mathbb{Z}}_2$ for the sign — any single transposition splits the sign homomorphism (see permutation-groups.md).
• Euclidean group $\mathrm{ISE}(n)={\mathbb{R}}^n⋊O(n)$: rotations and reflections act on translations. Rigid motions of ${\mathbb{R}}^n$.
• Poincaré group $\mathcal{P}={\mathbb{R}}^{1,3}⋊\mathcal{L}$: Lorentz transformations act on spacetime translations, ${\varphi }_{\Lambda }(a)=\Lambda a$, so $$({\Lambda }_1,a_1)\cdot ({\Lambda }_2,a_2)=({\Lambda }_1{\Lambda }_2,a_1+{\Lambda }_1{a}_2).$$ Translating then boosting differs from boosting then translating — the translation vector itself gets rotated. This is the structure behind Wigner's classification in QFT/preliminaries.md.
• Affine group $\mathrm{Aff}(n,\mathbb{F})={\mathbb{F}}^n⋊GL(n,\mathbb{F})$.

References

• Dummit & Foote, Abstract Algebra, §5.4–5.5.
• Rotman, An Introduction to the Theory of Groups, Ch. 7 (extensions and cohomology).