Unit 5: Group Homomorphisms, Isomorphism Theorems & Automorphisms

1. Definition of a Group Homomorphism

Let $(G, \cdot)$ and $(H, *)$ be two groups.

Definition 5.1 (Homomorphism): A function $\phi: G \to H$ is called a group homomorphism if it preserves the group operation:

• If $\phi$ is injective (one-to-one), it is called a monomorphism (or embedding).
• If $\phi$ is surjective (onto), it is called an epimorphism.
• If $\phi$ is bijective (both injective and surjective), it is called an isomorphism, denoted $G \cong H$.
• An isomorphism from $G$ to itself ($\phi: G \to G$) is called an automorphism.
• A homomorphism from $G$ to itself is called an endomorphism.

2. Fundamental Properties of Homomorphisms

Proposition 5.1 (Elementary Properties): Let $\phi: G \to H$ be a homomorphism. Then:

1. $\phi(e_G) = e_H$.
2. $\phi(g^{-1}) = (\phi(g))^{-1}$ for all $g \in G$.
3. $\phi(g^n) = (\phi(g))^n$ for all $g \in G$ and $n \in \mathbb{Z}$.
4. If $|g|$ is finite, then $|\phi(g)|$ divides $|g|$.
5. If $K \le G$, then $\phi(K) \le H$.
6. If $L \le H$, then the preimage $\phi^{-1}(L) \le G$.

Proof of 1, 2, and 4:

• Proof of 1: $\phi(e_G) = \phi(e_G \cdot e_G) = \phi(e_G) * \phi(e_G)$.

Multiplying both sides by $(\phi(e_G))^{-1}$ in $H$:

• Proof of 2: $\phi(g) * \phi(g^{-1}) = \phi(g \cdot g^{-1}) = \phi(e_G) = e_H$.

By uniqueness of inverses in $H$, $\phi(g^{-1}) = (\phi(g))^{-1}$. $\blacksquare$

• Proof of 4: Let $n = |g|$. Then $g^n = e_G$.

Applying $\phi$: $(\phi(g))^n = \phi(g^n) = \phi(e_G) = e_H$. By Proposition 2.3, this implies $|\phi(g)|$ divides $n = |g|$. $\blacksquare$

3. Kernel and Image

Definition 5.2 (Kernel and Image): Let $\phi: G \to H$ be a group homomorphism.

1. The kernel of $\phi$ is:

1. The image of $\phi$ is:

Theorem 5.1 (Kernel is a Normal Subgroup): Let $\phi: G \to H$ be a group homomorphism. Then:

1. $\ker(\phi) \trianglelefteq G$.
2. $\text{im}(\phi) \le H$.
3. $\phi$ is injective if and only if $\ker(\phi) = \{e_G\}$.

Complete Proof:

1. Normality of $\ker(\phi)$:

• Non-emptiness: $\phi(e_G) = e_H \implies e_G \in \ker(\phi)$.
• Subgroup test: Let $x, y \in \ker(\phi)$. Then:

Thus $x y^{-1} \in \ker(\phi)$, so $\ker(\phi) \le G$.

• Normality: For any $g \in G$ and $k \in \ker(\phi)$:

Hence $g k g^{-1} \in \ker(\phi)$ for all $g \in G$. Therefore, $\ker(\phi) \trianglelefteq G$. $\blacksquare$

2. Subgroup property of $\text{im}(\phi)$:

Let $u, v \in \text{im}(\phi)$. Then $u = \phi(a)$ and $v = \phi(b)$ for some $a, b \in G$.

Thus $\text{im}(\phi) \le H$. $\blacksquare$

3. Injectivity Criterion:

• $(\implies)$ If $\phi$ is injective and $\phi(x) = e_H = \phi(e_G)$, then injectivity implies $x = e_G$. Thus $\ker(\phi) = \{e_G\}$.
• $(\impliedby)$ Suppose $\ker(\phi) = \{e_G\}$. If $\phi(a) = \phi(b)$, then:

Thus $a b^{-1} \in \ker(\phi) = \{e_G\}$, meaning $a b^{-1} = e_G \implies a = b$. Therefore $\phi$ is injective. $\blacksquare$

1. Statement of the First Isomorphism Theorem

The First Isomorphism Theorem (often called the Fundamental Theorem of Homomorphisms) reveals that every homomorphism $\phi: G \to H$ can be factored into a canonical projection onto a quotient group followed by an isomorphism onto its image.

Theorem 5.2 (First Isomorphism Theorem): Let $\phi: G \to H$ be a group homomorphism with kernel $K = \ker(\phi)$. Then the quotient group $G/K$ is isomorphic to the image $\text{im}(\phi)$:

Specifically, the mapping:

is a well-defined group isomorphism.

``` G ------------ \phi ------------> H | ^ | \pi | v | \Phi G/K ------------ \cong -----------> im(\phi) ```

2. Complete Rigorous Proof

We must verify four assertions:

1. $\Phi$ is well-defined (independent of coset representative).
2. $\Phi$ is a homomorphism.
3. $\Phi$ is injective.
4. $\Phi$ is surjective.

Step 1: Well-Definedness

Suppose $g_1 K = g_2 K$ for $g_1, g_2 \in G$. Then $g_1^{-1} g_2 \in K = \ker(\phi)$. This means $\phi(g_1^{-1} g_2) = e_H$. Since $\phi$ is a homomorphism:

Therefore, $\Phi(g_1 K) = \phi(g_1) = \phi(g_2) = \Phi(g_2 K)$. The value of $\Phi(gK)$ is completely independent of the representative $g$.

Step 2: Homomorphism Property

For any cosets $aK, bK \in G/K$:

Since $\phi$ is a homomorphism:

Thus $\Phi$ preserves the group operation.

Step 3: Injectivity

We examine the kernel of $\Phi$:

Now $\Phi(gK) = e_H \iff \phi(g) = e_H \iff g \in \ker(\phi) = K$. By coset equality (Lemma 4.1), $g \in K \iff gK = K = e_{G/K}$. Therefore:

By Theorem 5.1, $\ker(\Phi) = \{ e_{G/K} \}$ implies $\Phi$ is injective.

Step 4: Surjectivity

Let $y \in \text{im}(\phi)$. By definition of the image, there exists some $g \in G$ such that $\phi(g) = y$. Then for the coset $gK \in G/K$, we have:

Thus every element of $\text{im}(\phi)$ is hit by $\Phi$. $\Phi$ is surjective.

Conclusion: $\Phi$ is a bijective homomorphism, hence an isomorphism:

1. The Second Isomorphism Theorem (Diamond Isomorphism Theorem)

The Second Isomorphism Theorem describes the relationship between the intersection and product of a subgroup and a normal subgroup.

Theorem 5.3 (Second Isomorphism Theorem): Let $G$ be a group, $H \le G$ a subgroup, and $N \trianglelefteq G$ a normal subgroup. Then:

1. $HN = \{ hn : h \in H, n \in N \} \le G$.
2. $N \trianglelefteq HN$.
3. $H \cap N \trianglelefteq H$.
4. The quotient groups are isomorphic:

``` HN / \ H N \ / H \cap N ```

Complete Proof:

1. $HN \le G$: Let $h_1 n_1, h_2 n_2 \in HN$. Then:

Since $N \trianglelefteq G$, $h_2^{-1} N h_2 = N$, so there exists $n_3 \in N$ such that $(n_1 n_2^{-1}) h_2^{-1} = h_2^{-1} n_3$. Then:

By the one-step subgroup test, $HN \le G$.

2. $N \trianglelefteq HN$: For any $hn \in HN$ and $m \in N$:

Since $N \trianglelefteq G$, $n m n^{-1} \in N$, and $h (n m n^{-1}) h^{-1} \in N$. Thus $N \trianglelefteq HN$.

3. Application of First Isomorphism Theorem:

Define the map:

• Homomorphism: $\theta(h_1 h_2) = (h_1 h_2) N = (h_1 N)(h_2 N) = \theta(h_1) \theta(h_2)$.
• Surjectivity: Any element of $HN/N$ is a coset of the form $(hn)N = h(nN) = hN$ (since $n \in N \implies nN = N$).

Thus $\theta(h) = hN = (hn)N$, so $\theta$ is surjective!

• Kernel:

By Theorem 5.1, the kernel of any homomorphism is normal in the domain, so $H \cap N \trianglelefteq H$. Applying the First Isomorphism Theorem (Theorem 5.2) to $\theta$:

2. The Third Isomorphism Theorem

Theorem 5.4 (Third Isomorphism Theorem): Let $G$ be a group, and let $N$ and $K$ be normal subgroups of $G$ such that $N \le K \trianglelefteq G$. Then:

1. $K/N \trianglelefteq G/N$.
2. The quotient of quotients satisfies:

Complete Proof:

Define the map $\psi: G/N \to G/K$ by:

• Well-definedness: If $g_1 N = g_2 N$, then $g_1^{-1} g_2 \in N$. Since $N \subseteq K$, $g_1^{-1} g_2 \in K$, which implies $g_1 K = g_2 K$.
• Homomorphism: $\psi((aN)(bN)) = \psi((ab)N) = (ab)K = (aK)(bK) = \psi(aN) \psi(bN)$.
• Surjectivity: For any coset $gK \in G/K$, $\psi(gN) = gK$.
• Kernel:

Applying the First Isomorphism Theorem to $\psi$:

1. The Concrete Universality of Symmetric Groups

Can every abstract group—no matter how exotic—be viewed as a group of permutations of some set? Arthur Cayley answered this affirmatively in 1854.

Theorem 5.5 (Cayley's Theorem): Every group $G$ is isomorphic to a subgroup of the symmetric group $\text{Sym}(G)$. In particular, if $G$ is a finite group of order $n$, then $G$ is isomorphic to a subgroup of $S_n$.

2. Complete Constructive Proof

For each element $g \in G$, define the left-multiplication map:

Step 1: $L_g$ is a Permutation of $G$

We show $L_g \in \text{Sym}(G)$ by showing it is bijective:

• Injective: If $L_g(x) = L_g(y)$, then $gx = gy$. Multiplying by $g^{-1}$ on the left gives $x = y$.
• Surjective: For any $y \in G$, $L_g(g^{-1}y) = g(g^{-1}y) = y$.

Thus $L_g: G \to G$ is a bijection, meaning $L_g \in \text{Sym}(G)$.

Step 2: The Left Regular Representation

Now define the mapping:

We prove that $\phi$ is a group homomorphism: For any $a, b \in G$ and any $x \in G$:

Since this holds for all $x \in G$:

Therefore, $\phi$ is a homomorphism.

Step 3: $\phi$ is Injective

We examine the kernel of $\phi$:

If $L_g = \text{id}_G$, then $L_g(x) = x$ for all $x \in G$. Evaluating at $x = e$:

Thus $\ker(\phi) = \{e\}$. By Theorem 5.1, $\phi$ is injective.

Step 4: Applying First Isomorphism Theorem

By the First Isomorphism Theorem (Theorem 5.2):

Therefore, $G$ is isomorphic to the subgroup $\text{im}(\phi) = \{ L_g : g \in G \} \le \text{Sym}(G)$. $\blacksquare$

1. The Automorphism Group $\text{Aut}(G)$

An automorphism of $G$ is an isomorphism $\alpha: G \to G$.

Theorem 5.6 (The Group $\text{Aut}(G)$): Under functional composition $\circ$, the set of all automorphisms of $G$, denoted $\text{Aut}(G)$, forms a group.

• Identity: $\text{id}_G(x) = x$.
• Inverse: The functional inverse $\alpha^{-1}$ of an isomorphism is an isomorphism.
• Associativity: Inherited from composition of functions.

2. Inner Automorphisms $\text{Inn}(G)$

For any fixed element $g \in G$, define the conjugation map by $g$:

Lemma 5.2: $\gamma_g$ is an automorphism of $G$, called the inner automorphism induced by $g$.

Proof:

• Homomorphism: $\gamma_g(xy) = g(xy)g^{-1} = (gxg^{-1})(gyg^{-1}) = \gamma_g(x) \gamma_g(y)$.
• Bijective: $\gamma_g \circ \gamma_{g^{-1}} = \gamma_{g^{-1}} \circ \gamma_g = \text{id}_G$. Thus $\gamma_g \in \text{Aut}(G)$. $\blacksquare$

The set of all inner automorphisms is denoted $\text{Inn}(G) = \{ \gamma_g : g \in G \}$.

Theorem 5.7 ($\text{Inn}(G) \cong G / Z(G)$):

1. $\text{Inn}(G)$ is a normal subgroup of $\text{Aut}(G)$ ($\text{Inn}(G) \trianglelefteq \text{Aut}(G)$).
2. $\text{Inn}(G) \cong G / Z(G)$, where $Z(G)$ is the center of $G$.

Complete Proof:

Define the map $\Gamma: G \to \text{Aut}(G)$ by $\Gamma(g) = \gamma_g$.

1. Homomorphism: For any $a, b \in G$ and $x \in G$:

Thus $\Gamma(ab) = \Gamma(a) \circ \Gamma(b)$.

2. Kernel of $\Gamma$:

3. Application of First Isomorphism Theorem:

The image of $\Gamma$ is by definition $\text{Inn}(G)$. By Theorem 5.2:

3. Outer Automorphisms & Characteristic Subgroups

Definition 5.3 (Outer Automorphism Group): The quotient group:

is called the outer automorphism group of $G$. Elements of $\text{Out}(G)$ are cosets of $\text{Inn}(G)$ called outer automorphism classes.

Definition 5.4 (Characteristic Subgroup): A subgroup $H \le G$ is called characteristic in $G$ (denoted $H \text{ char } G$) if for every $\alpha \in \text{Aut}(G)$:

Key Distinctions:

• Normal: Invariant under inner automorphisms ($\gamma_g(N) = N$ for all $g \in G$).
• Characteristic: Invariant under all automorphisms.
• Every characteristic subgroup is normal ($H \text{ char } G \implies H \trianglelefteq G$).
• Transitivity: While normality is NOT transitive in general ($K \trianglelefteq H \trianglelefteq G \not\implies K \trianglelefteq G$), characteristic subgroups ARE transitive: