Group homomorphism

Group monomorphism

Let 𝜑 :𝐺 →𝐻 be a group homomorphism. The following statements are equivalent: group

• 𝜑 is injective;
• 𝜑 is monic;
• ker⁡𝜑 =1

Proof

𝜑 is injective iff 𝜑(𝑎) =𝜑(𝑏) ⟹ 𝑎 =𝑏 iff 𝜑(𝑎𝑏−1) =𝑒 ⟹ 𝑎𝑏−1 =𝑒 iff ker⁡𝜑 ={𝑒}. Clearly injective 𝜑 implies monic 𝜑. Now let 𝜑 :𝐺 ↣𝐻 be a monomorphism. Let 𝜄 :ker⁡𝜑 ↪𝐺 be the canonical injection and 𝛼𝑇 :ker⁡𝜑 →𝐺 :𝑥 ↦𝑒 be the trivial homomorphism. Since 𝜑𝜄 =𝜑𝛼𝑇, 𝜄 =𝛼𝑇 and hence ker⁡𝜑 ={𝑒}.

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