The Kernel of a Group Homomorphism – Abstract Algebra

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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 • ♦♦♦♦♦♦♦♦♦♦

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