Homomorphism Summary

Abstract Algebra · Axiom Academy

SUMMARY Group Homomorphisms and Quotient Groups Let's review how homomorphisms preserve structure, connect groups through kernels and images, and enable quotient group construction via the fundamental isomorphism theorems. Definition: A function φ: G → H between groups satisfying φ(ab) = φ(a)φ(b) for all a, b ∈ G Structure Preservation: Products in G map to products in H—the operation structure is preserved Key Properties: Identity maps to identity: φ(e G ) = e H . Inverses map to inverses: φ(a⁻¹) = φ(a)⁻¹ Common Examples: Determinant det: GL n (ℝ) → ℝ*, sign map sgn: S n → ±1 , reduction mod n: ℤ → ℤ n Kernel: Ker(φ) = g ∈ G : φ(g) = e H measures what gets "collapsed" to the identity Image: Im(φ) = φ(g) : g ∈ G is the range of the homomorphism Subgroup Properties: Ker(φ) is always a normal subgroup of G. Im(φ) is always a subgroup of H Injectivity Test: φ is injective (one-to-one) if and only if Ker(φ) = e G Definition: N is normal in G (written N ⊴ G ) if gNg⁻¹ = N for all g ∈ G Equivalent Conditions: Left cosets equal right cosets ( gN = Ng ), or N is the kernel of some homomorphism Why It Matters: Normal subgroups are precisely those for which coset multiplication is well-defined Key Examples: All subgroups of abelian groups are normal. A n ⊴ S n . Kernels of homomorphisms are always normal Construction: Given N ⊴ G , the quotient group G/N consists of cosets with operation (aN)(bN) = (ab)N Order Formula: |G/N| = |G|/|N| counts the number of distinct cosets

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