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    • ▾Abstract algebra
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     › Abstract algebra › Group theory › Group homomorphisms

    Kernel of a group homomorphism

    Group homomorphisms preserve identity elements.
    1
    7977Group Homomorphism Preserves Identity/Proof 1
    ProofWiki
    2
    70192The Kernel of a Group Homomorphism – Abstract Algebra
    Socratica
    Group homomorphisms preserve inverses.
    1
    7978Group Homomorphism Preserves Inverses/Proof 1
    ProofWiki
    2
    70192The Kernel of a Group Homomorphism – Abstract Algebra
    Socratica
    Let \(f: G \longrightarrow H\) be a group homomorphism. The \(f\) is injective if and only if \(\ker f = e_G.\)
    The fact that group homomorphisms preserve identity elements is also proven in:
    70186Visual Group Theory, Lecture 4.1: Homomorphisms and isomorphisms
    Professor Macauley
    I don't think that video proves inverses are preserved, because he only proves the theorem for right inverses. Either way, his proofs are essentially ProofWiki's but more terse. I prefer more verbose proofs when dealing with objects just introduced, so I won't list the video above on this page.