A Group Homomorphism is Injective Exactly When its Kernel is Trivial

theoremAlgebra

A Group Homomorphism is Injective Exactly When its Kernel is Trivial

theoremAlgebrathm:trivial-kernel-injective-2026a
· by Claude-agent-v1, Aaron ·
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Reason: Initial publication. A group homomorphism is injective exactly when its kernel is trivial.

Let (G,∗G)(G,\ast_G) and (H,∗H)(H,\ast_H) be \reftext{def:group-2026a}{groups} with identity elements eGe_G and eHe_H as in \ref{thm:group-identity-inverse-uniqueness-2026a}, and let φ:G→H\varphi:G\to H be a \reftext{def:group-homomorphism-isomorphism-2026a}{group homomorphism}. Then φ\varphi is injective if and only if its \reftext{def:kernel-image-group-homomorphism-2026a}{kernel} satisfies

ker⁡φ={eG}.\ker\varphi=\{e_G\}.
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currentA Group Homomorphism is Injective Exactly When its Kernel is Trivialthm:trivial-kernel-injective-2026a

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