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What is a Group Homomorphism Definition and Example Abstract Algebra
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We introduce group homomorphisms with the definition, the definition of a homomorphic image, and several examples, including proof that certain functions are homomorphisms. We also show a non-example, a function from one group to another which is not a homomorphism. In short, a homomorphism f from a group G to a group H, is a function from G to H such that for all a, b in G, f(ab) = f(a)f(b). We say the homomorphism preserves the group operation . #abstractalgebra #grouptheory • Group Isomorphisms: • Isomorphic Groups and Isomorphisms in... • Basic Properties of Homomorphisms: • Proof: Basic Properties of Homomorphi... • Examples of Group Homomorphisms: coming soon • Kernels of Homomorphisms: • Kernels of Homomorphisms | Abstract A... • Abstract Algebra Course: • Abstract Algebra • Abstract Algebra Exercises: • Abstract Algebra Exercises • 0:00 - What is a Homomorphism? • 0:40 - Definition of a Homomorphism • 2:59 - Definition of Homomorphic Image • 4:30 - Example 1 (A homomorphism that is onto) • 10:09 - Example 2 (A homomorphism that is not onto) • 11:42 - Trivial Examples of Homomorphisms • 12:25 - Non-Example (A function that is not a homomorphism) • 13:03 - Two Basic Properties of Homomorphisms • ◉Textbooks I Like◉ • Graph Theory: https://amzn.to/3JHQtZj • Real Analysis: https://amzn.to/3CMdgjI • Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) • Statistics: https://amzn.to/3tsaEER • Abstract Algebra: https://amzn.to/3IjoZaO • Discrete Math: https://amzn.to/3qfhoUn • Number Theory: https://amzn.to/3JqpOQd • ★DONATE★ • ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons • ◆ Donate on PayPal: https://www.paypal.me/wrathofmath • Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon! • Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ • Follow Wrath of Math on... • ● Instagram: / wrathofmathedu • ● Facebook: / wrathofmath • ● Twitter: / wrathofmathedu • My Math Rap channel: / @mathbars2020
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