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The kernel of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism f:G → H. The kernel is the set of all elements in G which map to the identity element in H. It is a subgroup in G and it depends on f. Different homomorphisms between G and H can give different kernels. If f is an isomorphism, then the kernel will simply be the identity element. You can also define a kernel for a homomorphism between other objects in abstract algebra: rings, fields, vector spaces, modules. We will cover these in separate videos. Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ We recommend the following textbooks: Dummit & Foote, Abstract Algebra 3rd Edition http://amzn.to/2oOBd5S Milne, Algebra Course Notes (available free online) http://www.jmilne.org/math/CourseNote... ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : / socratica ► Make a one-time PayPal donation: https://www.paypal.me/socratica ► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9 Thank you! ♦♦♦♦♦♦♦♦♦♦ Connect with us! Facebook: / socraticastudios Instagram: / socraticastudios Twitter: / socratica ♦♦♦♦♦♦♦♦♦♦ Teaching Assistant: Liliana de Castro Written & Directed by Michael Harrison Produced by Kimberly Hatch Harrison ♦♦♦♦♦♦♦♦♦♦