Cool Examples of Quotient Groups | Abstract Algebra

We introduce several interesting examples of quotient groups in abstract algebra. In our first example we consider the normal subgroup H of an abelian group G which consists of all finite order elements. Then, in taking the quotient group G/H, we see that we factor out all the finite order elements, leaving in G/H only the identity as a finite order element. In our second example we consider the normal subgroup H of all commutators of a group G. We'll see in the quotient group G/H, we have in a sense factored out all the commutators, leaving only the trivial identity behind, and hence having G/H as an abelian group. We finish with an example regarding the center of a group G. Note that quotient groups are sometimes also called factor groups. #abstractalgebra Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join Quotient Groups: https://youtu.be/MKDAZV0XasI Coset Property we Used in Proof: https://youtu.be/v9Hi-xNGO2c Abstract Algebra Course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra Exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz Get the textbook for this course! https://amzn.to/3IjoZaO 0:00 Intro 0:10 Example 1 6:27 Example 2 9:59 Example 3 14:21 Conclusion ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Outro music by Ben Watts and is available for channel members. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath