Trivial Homomorphic Images in Group Theory
An exploration of the trivial homomorphic images of groups, demonstrating that every group has exactly two such images: the trivial group and the group itself. The proof is straightforward and fundamental to understanding homomorphisms in abstract algebra

Wrath of Math
725 views • Sep 20, 2023

About this video
We prove every group has two trivial homomorphic images, the trivial group with only the identity, and the whole group itself. The proof is straightforward, requiring only the knowledge of the definition of a homomorphism and a homomorphic image, which relates to surjectivity surjectivity or a function being "onto". #abstractalgebra #grouptheory
Group Homomorphisms: https://youtu.be/rJpu22jMeIY
Basic Properties of Group Homomorphisms: https://youtu.be/7tRDyAbgrJ8
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Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz
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Group Homomorphisms: https://youtu.be/rJpu22jMeIY
Basic Properties of Group Homomorphisms: https://youtu.be/7tRDyAbgrJ8
Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN
Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz
◉Textbooks I Like◉
Graph Theory: https://amzn.to/3JHQtZj
Real Analysis: https://amzn.to/3CMdgjI
Abstract Algebra: https://amzn.to/3IjoZaO
Linear Algebra: https://amzn.to/43xAWEz
Calculus: https://amzn.to/3PieD1M
Proofs and Set Theory: https://amzn.to/367VBXP (available for free online)
Statistics: https://amzn.to/3tsaEER
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: https://www.patreon.com/join/wrathofmathlessons
◆ Donate on PayPal: https://www.paypal.me/wrathofmath
Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!
Outro music is mine. You cannot find it anywhere, for now.
Follow Wrath of Math on...
● Instagram: https://www.instagram.com/wrathofmathedu
● Facebook: https://www.facebook.com/WrathofMath
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Video Information
Views
725
Likes
19
Duration
3:42
Published
Sep 20, 2023
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