⭕Groups and Geometries Unit 2 – Group Homomorphisms & Isomorphisms

Key Concepts

• Group homomorphisms map elements from one group to another while preserving the group operation
• Isomorphisms are bijective homomorphisms that establish a one-to-one correspondence between two groups
• Kernels consist of all elements in the domain group that map to the identity element in the codomain group
• Images refer to the subset of elements in the codomain group that are mapped to by elements in the domain group
• Automorphisms are isomorphisms from a group to itself
• Theorems such as the First Isomorphism Theorem relate homomorphisms, kernels, and quotient groups
• Group homomorphisms and isomorphisms have applications in geometry, such as studying symmetries and transformations

Definition and Properties

• A group homomorphism is a function $f: G \rightarrow H$ between two groups $(G, *)$ and $(H, \circ)$ that satisfies the homomorphism property: $f(a * b) = f(a) \circ f(b)$ for all $a, b \in G$
• Homomorphisms preserve the group structure, meaning they map the identity element of $G$ to the identity element of $H$ and map inverses in $G$ to inverses in $H$
• The composition of two group homomorphisms is also a group homomorphism
• If a group homomorphism is bijective (both injective and surjective), it is called an isomorphism
• Isomorphic groups have the same group structure and properties, differing only in the notation of their elements

Types of Group Homomorphisms

• Monomorphisms (injective homomorphisms) are one-to-one, meaning distinct elements in the domain group map to distinct elements in the codomain group
  • Example: The inclusion map from the integers $(\mathbb{Z}, +)$ to the rational numbers $(\mathbb{Q}, +)$ is a monomorphism
• Epimorphisms (surjective homomorphisms) map the domain group onto the entire codomain group, meaning every element in the codomain group has at least one preimage in the domain group
  • Example: The quotient map from the integers $(\mathbb{Z}, +)$ to the quotient group $\mathbb{Z}/n\mathbb{Z}$ (integers modulo $n$) is an epimorphism
• Endomorphisms are homomorphisms from a group to itself, i.e., the domain and codomain groups are the same
• Isomorphisms are bijective homomorphisms that establish a one-to-one correspondence between two groups, preserving the group structure in both directions
• Automorphisms are isomorphisms from a group to itself, forming the symmetries of the group

Kernels and Images

• The kernel of a group homomorphism $f: G \rightarrow H$ is the set of all elements in $G$ that map to the identity element $e_H$ in $H$: $\ker(f) = \{g \in G \mid f(g) = e_H\}$
• The kernel is always a normal subgroup of the domain group $G$
• The image of a group homomorphism $f: G \rightarrow H$ is the set of all elements in $H$ that are mapped to by elements in $G$: $\operatorname{im}(f) = \{f(g) \mid g \in G\}$
• The image is always a subgroup of the codomain group $H$
• The First Isomorphism Theorem states that for a group homomorphism $f: G \rightarrow H$, the quotient group $G/\ker(f)$ is isomorphic to the image $\operatorname{im}(f)$

Isomorphisms and Automorphisms

• An isomorphism is a bijective group homomorphism that establishes a one-to-one correspondence between two groups
• Isomorphic groups have the same group structure and properties, differing only in the notation of their elements
• The inverse of an isomorphism is also an isomorphism
• Automorphisms are isomorphisms from a group to itself, forming the symmetries of the group
• The set of all automorphisms of a group $G$, denoted $\operatorname{Aut}(G)$, forms a group under function composition
• Inner automorphisms are automorphisms induced by conjugation with elements of the group, while outer automorphisms cannot be expressed as conjugation

Theorems and Proofs

• The First Isomorphism Theorem (Fundamental Theorem of Group Homomorphisms) states that for a group homomorphism $f: G \rightarrow H$, the quotient group $G/\ker(f)$ is isomorphic to the image $\operatorname{im}(f)$
  • This theorem relates the kernel, image, and quotient groups of a homomorphism
• The proof of the First Isomorphism Theorem involves constructing a well-defined, bijective homomorphism between $G/\ker(f)$ and $\operatorname{im}(f)$
• The Second Isomorphism Theorem states that if $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, then $H/(H \cap N)$ is isomorphic to $(HN)/N$
• The Third Isomorphism Theorem states that if $G$ is a group and $N, M$ are normal subgroups of $G$ with $N \subseteq M$, then $(G/N)/(M/N)$ is isomorphic to $G/M$
• Cayley's Theorem states that every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$, denoted $S_G$

Applications in Geometry

• Group homomorphisms and isomorphisms are used to study symmetries and transformations in geometry
• The symmetry group of a geometric object consists of all transformations that leave the object invariant
  • Example: The symmetry group of a square includes rotations by multiples of 90 degrees and reflections about its axes of symmetry
• Isomorphisms between symmetry groups indicate that the corresponding geometric objects have the same symmetry properties
• The Euclidean group, denoted $E(n)$, is the group of isometries (distance-preserving transformations) of $n$-dimensional Euclidean space
  • Translations, rotations, and reflections are examples of isometries in Euclidean space
• The study of crystallographic groups, which describe the symmetries of crystal lattices, relies on group homomorphisms and isomorphisms

Practice Problems and Examples

1. Determine whether the function $f: (\mathbb{R}, +) \rightarrow (\mathbb{R}^+, \cdot)$ defined by $f(x) = e^x$ is a group homomorphism.
2. Prove that the kernel of a group homomorphism is always a normal subgroup of the domain group.
3. Find the automorphism group of the cyclic group $\mathbb{Z}_6$.
4. Show that the groups $(\mathbb{Z}_4, +)$ and $(\{1, -1, i, -i\}, \cdot)$ are isomorphic.
5. Describe the symmetry group of a regular hexagon and find its order.
6. Apply the First Isomorphism Theorem to the homomorphism $f: (\mathbb{Z}, +) \rightarrow (\mathbb{Z}_6, +)$ defined by $f(x) = [x]_6$ (the equivalence class of $x$ modulo 6).
7. Prove that the composition of two group isomorphisms is also an isomorphism.
8. Determine the kernel and image of the homomorphism $f: (\mathbb{Z}_12, +) \rightarrow (\mathbb{Z}_6, +)$ defined by $f([x]_{12}) = [x]_6$.