Free Products of Groups

Algebraic Topology · Axiom Academy

LESSON Free Products of Groups Combining groups without imposing relations between their elements Given two groups G and H, their free product G * H consists of all finite sequences (words) formed by alternating elements from G and H, where consecutive elements come from different groups. Formal Definition: Elements of G * H are reduced words of the form: where gᵢ ∈ G \ e_G and hⱼ ∈ H \ e_H , with group operation being concatenation followed by reduction. The free product satisfies a universal property: any pair of homomorphisms φ: G → K and ψ: H → K extends uniquely to a homomorphism Φ: G * H → K. Universal Property: For any group K with homomorphisms φ: G → K and ψ: H → K, there exists a unique homomorphism Φ: G * H → K such that: If G and H are presented by generators and relations, we can construct a presentation for G * H by combining their presentations. If G = ⟨S_G | R_G⟩ and H = ⟨S_H | R_H⟩, then: Let's examine concrete examples to understand the structure of free products. Example 1: ℤ * ℤ is the free group on two generators. Example 2: ℤ/2ℤ * ℤ/2ℤ contains elements like aba⁻¹b⁻¹ of infinite order. Example 3: If G = ⟨a | a² = e⟩ and H = ⟨b | b³ = e⟩, then G * H = ⟨a, b | a² = e, b³ = e⟩.

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