Group Homomorphism

About these notes: These notes are based on Abstract Algebra: An Integrated Approach (Joseph H. Silverman). Mathematical facts are not copyrighted, but these notes represent my own original summarization and explanation. Any errors or typos are entirely my own.

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1. Definition (Group Homomorphism)

Let \(G\) and \(G'\) be groups. A function \(\phi: G \to G'\) is a group homomorphism 1 Abstract Algebra: An Integrated Approach (Silverman), 2.24, page 44. if it preserves the group operation: \[ \phi(g_1 \cdot g_2) = \phi(g_1) \cdot \phi(g_2) \qquad \forall g_1, g_2 \in G. \]

Intuition: A homomorphism is a “structure-preserving” map. It doesn’t matter if you combine two elements in \(G\) first and then map the result, or if you map the elements to \(G'\) first and then combine them. The result is the same.

2. Basic Properties of Homomorphisms

These properties are not additional requirements; they are automatic consequences of the single definition \(\phi(g_1 \cdot g_2) = \phi(g_1) \cdot \phi(g_2)\).

2.1. Proposition (Preservation of Identity)

Let \(e_G\) and \(e_{G'}\) be the identity elements of \(G\) and \(G'\), respectively. Then \(\phi(e_G) = e_{G'}\).

2.2. Proposition (Preservation of Inverses)

For every \(g \in G\), \(\phi(g^{-1}) = (\phi(g))^{-1}\).

2.3. Proposition (Preservation of Powers)

For every \(g \in G\) and every integer \(n \in \mathbb{Z}\), \(\phi(g^n) = (\phi(g))^n\).

3. Kernel and Image

Two fundamental subsets associated with any homomorphism \(\phi: G \to G'\).

3.1. Definition (Kernel)

The kernel of \(\phi\), denoted \(\ker(\phi)\), is the set of all elements in \(G\) that map to the identity element of \(G'\): \[ \ker(\phi) = \{g \in G \mid \phi(g) = e_{G'}\}. \]

3.2. Definition (Image)

The image of \(\phi\), denoted \(\text{Im}(\phi)\) or \(\phi(G)\), is the set of all actual outputs in \(G'\): \[ \text{Im}(\phi) = \{\phi(g) \mid g \in G\}. \]

3.3. Proposition (Subgroup Property)

\(\ker(\phi)\) is always a subgroup of \(G\), and \(\text{Im}(\phi)\) is always a subgroup of \(G'\).

4. Types of Homomorphisms

Depending on the set-theoretic properties of the function \(\phi\), we give homomorphisms special names:

  1. Monomorphism (Injective / One-to-One): No two elements in \(G\) map to the same element in \(G'\).
    • Shortcut: \(\phi\) is injective \(\iff \ker(\phi) = \{e_G\}\).
  2. Epimorphism (Surjective / Onto): Every element in \(G'\) is hit by at least one element in \(G\).
    • Shortcut: \(\phi\) is surjective \(\iff \text{Im}(\phi) = G'\).
  3. Isomorphism (Bijective): \(\phi\) is both injective and surjective2 Abstract Algebra: An Integrated Approach (Silverman), 2.30, page 45. . If an isomorphism exists between \(G\) and \(G'\), we write \(G \cong G'\) and say the groups are isomorphic. 3 Some texts use the term “isomorphism” to refer to the map itself, and “isomorphic” to describe the relationship between the two groups.

5. Structural Properties of Isomorphic Groups

Why this matters: Isomorphism is the strongest notion of “sameness” in mathematics. Isomorphic groups are literally the same group wearing different costumes (different names for elements or operations).

Isomorphic groups have identical structural properties. If \(G_1 \cong G_2\) via an isomorphism \(\phi\), then:

  1. If one is finite, both are finite, and they satisfy \(\#G_1 = \#G_2\).
  2. If one is abelian, then both are abelian.
  3. If \(G_1\) has an element of order \(n\), then \(G_2\) also has an element of order \(n\) (specifically, \(\phi(g)\) has order \(n\)).
  4. More generally, if \(G_1\) has \(k\) different elements of order \(n\), then \(G_2\) also has exactly \(k\) different elements of order \(n\).

6. Examples

6.1. The Trivial Homomorphism

For any groups \(G\) and \(G'\), the function \(\phi(g) = e_{G'}\) for all \(g \in G\) is a homomorphism.

  • It is rarely injective (unless \(G\) is the trivial group).
  • Its kernel is all of \(G\), and its image is \(\{e_{G'}\}\).

6.2. The Standard Inclusion Map (\(S_m \to S_n\))

Let \(n \ge m \ge 1\). Define \(\phi: S_m \to S_n\) such that for any \(\pi \in S_m\):

  • \(\phi(\pi)(x) = \pi(x)\) for \(1 \le x \le m\)
  • \(\phi(\pi)(x) = x\) for \(m < x \le n\) 4 This map takes a permutation of \(m\) items and leaves the remaining \(n-m\) items fixed.
  • It is injective (a monomorphism) because \(\ker(\phi) = \{id_m\}\). No information is lost.
  • It is not surjective (if \(n > m\)) because permutations in \(S_n\) that move elements greater than \(m\) are never reached.

6.3. The Determinant Map

Let \(G = \text{GL}_n(\mathbb{R})\) under matrix multiplication, and \(G' = (\mathbb{R} \setminus \{0\}, \times)\). Define \(\phi(A) = \det(A)\).

  • Homomorphism: \(\det(AB) = \det(A)\det(B)\), so \(\phi(AB) = \phi(A)\phi(B)\).
  • Surjective: For any \(r \neq 0\), the diagonal matrix with \(r\) in the top-left and \(1\) s elsewhere maps to \(r\).
  • Not Injective: Many different matrices have the same determinant (e.g., any rotation matrix has determinant 1). The kernel is the Special Linear Group \(\text{SL}_n(\mathbb{R})\).

6.4. The Logarithm Isomorphism

Let \(G = (\mathbb{R}^+, \times)\) and \(G' = (\mathbb{R}, +)\). Define \(\phi(x) = \ln(x)\).

  • Homomorphism: \(\ln(x \cdot y) = \ln(x) + \ln(y)\).
  • Bijective: The natural logarithm is strictly increasing (injective) and its range is all real numbers (surjective).
  • Conclusion: \(\phi\) is an isomorphism. This is why logarithms were historically invented: to translate difficult multiplication problems into easy addition problems!