The Kernel of a Group Homomorphism – Abstract Algebra

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/CourseNotes/index.html

♦♦♦♦♦♦♦♦♦♦

Ways to support our channel:

► Join our Patreon : https://www.patreon.com/socratica
► Make a one-time PayPal donation: https://www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9

Thank you!

♦♦♦♦♦♦♦♦♦♦

Connect with us!

Facebook: https://www.facebook.com/SocraticaStudios/
Instagram: https://www.instagram.com/SocraticaStudios/
Twitter: https://twitter.com/Socratica

♦♦♦♦♦♦♦♦♦♦

Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison

♦♦♦♦♦♦♦♦♦♦

Видео The Kernel of a Group Homomorphism – Abstract Algebra канала Socratica