First Isomorphism Theorem

Abstract Algebra · Axiom Academy

LESSON First Isomorphism Theorem One of the most fundamental results in group theory: every homomorphism naturally creates an isomorphism between its quotient and image. More precisely: there exists an isomorphism ψ: G/Ker(φ) → Im(φ) defined by ψ(gKer(φ)) = φ(g). This map is well-defined, bijective, and preserves the group operation. 2. Understanding the Components Before proving the theorem, let's understand what each piece means: Kernel: Ker(φ) = g ∈ G : φ(g) = e H Image: Im(φ) = h ∈ H : h = φ(g) for some g ∈ G Quotient Group: G/Ker(φ) = gKer(φ) : g ∈ G The kernel partitions G into cosets, and the image is a subgroup of H. The theorem says these structures are fundamentally the same. The key insight: elements in the same coset of Ker(φ) all map to the same element in the image. This creates a perfect one-to-one correspondence. Consider the natural projection φ: ℤ → ℤₙ defined by φ(k) = [k mod n]. The isomorphism ψ: ℤ/nℤ → ℤₙ maps each coset [k] + nℤ to [k] ∈ ℤₙ. This shows that quotient groups and modular arithmetic are the same thing! 5. Example 2: det: GLₙ(ℝ) → ℝ* The determinant function det: GLₙ(ℝ) → ℝ* is a group homomorphism from invertible n×n matrices to nonzero real numbers under multiplication. The theorem gives us: GLₙ(ℝ)/SLₙ(ℝ) ≅ ℝ*. This means the "space of all volume scalings" is isomorphic to the nonzero reals! The sign homomorphism sign: Sₙ → ±1 maps each permutation to +1 if it's even (even number of transpositions) or -1 if it's odd.

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