⭕Groups and Geometries Unit 5 – Direct and Semidirect Products

Key Concepts

• Direct product combines two groups into a larger group where elements are ordered pairs and group operation is performed component-wise
• Semidirect product is a generalization of direct product that allows for "twisting" of one group by the other through a homomorphism
• Normal subgroups play a crucial role in constructing semidirect products as they allow for a well-defined group operation
• Isomorphisms and homomorphisms provide a way to relate direct and semidirect products to other known groups
• Geometric interpretations of direct and semidirect products offer insights into their structure and symmetries

Definition and Properties

• Direct product of groups $G$ and $H$, denoted $G \times H$, is the set of ordered pairs $(g, h)$ where $g \in G$ and $h \in H$
  • Group operation is defined component-wise: $(g_1, h_1) \cdot (g_2, h_2) = (g_1g_2, h_1h_2)$
  • Identity element is $(e_G, e_H)$, where $e_G$ and $e_H$ are identities of $G$ and $H$, respectively
  • Inverse of $(g, h)$ is $(g^{-1}, h^{-1})$
• Semidirect product of groups $N$ and $H$, denoted $N \rtimes_{\varphi} H$, requires a homomorphism $\varphi: H \to \text{Aut}(N)$
  • Elements are ordered pairs $(n, h)$ where $n \in N$ and $h \in H$
  • Group operation is defined as $(n_1, h_1) \cdot (n_2, h_2) = (n_1 \varphi(h_1)(n_2), h_1h_2)$
• Both direct and semidirect products satisfy group axioms (closure, associativity, identity, inverses)
• Direct product is commutative (abelian) if and only if both $G$ and $H$ are commutative
• Semidirect product is generally non-commutative, even if $N$ and $H$ are commutative

Construction Methods

• To construct a direct product $G \times H$, simply form ordered pairs $(g, h)$ and define the group operation component-wise
• Constructing a semidirect product $N \rtimes_{\varphi} H$ requires:
  1. A normal subgroup $N$ of a group $G$
  2. A subgroup $H$ of $G$ such that $G = NH$ and $N \cap H = \{e\}$
  3. A homomorphism $\varphi: H \to \text{Aut}(N)$ defined by $\varphi(h)(n) = hnh^{-1}$
• The resulting semidirect product has elements $(n, h)$ and group operation $(n_1, h_1) \cdot (n_2, h_2) = (n_1 \varphi(h_1)(n_2), h_1h_2)$
• External semidirect product can be constructed when $N$ and $H$ are given separately with a homomorphism $\varphi: H \to \text{Aut}(N)$
• Internal semidirect product arises from a group $G$ with a normal subgroup $N$ and a subgroup $H$ satisfying the conditions mentioned above

Examples and Applications

• Direct product of cyclic groups $\mathbb{Z}_m \times \mathbb{Z}_n$ has order $mn$ and is cyclic if and only if $m$ and $n$ are coprime
• Dihedral group $D_{2n}$ is a semidirect product of cyclic groups: $D_{2n} \cong \mathbb{Z}_n \rtimes_{\varphi} \mathbb{Z}_2$, where $\varphi(1)(k) = -k$
• Quaternion group $Q_8$ is a semidirect product: $Q_8 \cong \mathbb{Z}_4 \rtimes_{\varphi} \mathbb{Z}_2$, with $\varphi(1)(k) = -k$
• Euclidean group $E(2)$ of rigid motions in the plane is a semidirect product of translations and rotations: $E(2) \cong \mathbb{R}^2 \rtimes_{\varphi} SO(2)$
• Wreath products, such as the lamplighter group, can be constructed as semidirect products of simpler groups

Subgroups and Normal Subgroups

• Subgroups of a direct product $G \times H$ are of the form $K \times L$, where $K \leq G$ and $L \leq H$
• Normal subgroups of a direct product $G \times H$ are of the form $K \times L$, where $K \trianglelefteq G$ and $L \trianglelefteq H$
• In a semidirect product $N \rtimes_{\varphi} H$, $N \times \{e\}$ is always a normal subgroup isomorphic to $N$
• $\{e\} \times H$ is a subgroup of $N \rtimes_{\varphi} H$ isomorphic to $H$, but not necessarily normal
• Subgroups and normal subgroups of semidirect products can be more complex and depend on the specific homomorphism $\varphi$

Isomorphisms and Homomorphisms

• If $G \cong G'$ and $H \cong H'$, then $G \times H \cong G' \times H'$
• Isomorphism of semidirect products requires isomorphic normal subgroups, isomorphic acting groups, and compatible homomorphisms
• Projection maps $\pi_1: G \times H \to G$ and $\pi_2: G \times H \to H$ are homomorphisms
• In a semidirect product $N \rtimes_{\varphi} H$, the map $(n, h) \mapsto h$ is a homomorphism onto $H$ with kernel $N \times \{e\}$
• Homomorphisms between semidirect products can be constructed using homomorphisms of their constituent groups that respect the semidirect product structure

Geometric Interpretations

• Direct product of two cyclic groups $\mathbb{Z}_m \times \mathbb{Z}_n$ can be visualized as a discrete torus (grid on a torus surface)
• Semidirect product structure of dihedral groups $D_{2n}$ reflects the symmetries of regular $n$-gons
  • Rotations correspond to elements of $\mathbb{Z}_n$
  • Reflections correspond to elements of $\mathbb{Z}_2$ acting on $\mathbb{Z}_n$ by inversion
• Euclidean group $E(2)$ as a semidirect product captures the interplay between translations and rotations in the plane
  • Translations form a normal subgroup isomorphic to $\mathbb{R}^2$
  • Rotations act on translations by rotating the translation vectors
• Semidirect products can describe symmetries of various geometric objects, such as frieze patterns and wallpaper groups

Practice Problems and Solutions

1. Prove that the direct product of two cyclic groups $\mathbb{Z}_m \times \mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_{mn}$ if and only if $\gcd(m, n) = 1$.
   • Solution: Use the Chinese Remainder Theorem and the fundamental theorem of finitely generated abelian groups
2. Show that the quaternion group $Q_8$ is isomorphic to a semidirect product of $\mathbb{Z}_4$ and $\mathbb{Z}_2$.
   • Solution: Define $\varphi: \mathbb{Z}_2 \to \text{Aut}(\mathbb{Z}_4)$ by $\varphi(1)(k) = -k$ and show that $Q_8 \cong \mathbb{Z}_4 \rtimes_{\varphi} \mathbb{Z}_2$
3. Prove that a semidirect product $N \rtimes_{\varphi} H$ is a direct product if and only if $\varphi(h) = \text{id}_N$ for all $h \in H$.
   • Solution: Show that the group operation reduces to component-wise multiplication if and only if $\varphi$ is trivial
4. Find all non-isomorphic semidirect products of $\mathbb{Z}_3$ and $\mathbb{Z}_4$.
   • Solution: Determine all homomorphisms $\varphi: \mathbb{Z}_4 \to \text{Aut}(\mathbb{Z}_3)$ and construct the corresponding semidirect products
5. Prove that every group of order $pq$, where $p$ and $q$ are primes with $p \mid (q - 1)$, is a semidirect product of cyclic groups.
   • Solution: Use Sylow theorems and the classification of groups of order $pq$ to show that such groups are semidirect products of $\mathbb{Z}_p$ and $\mathbb{Z}_q$