Wrath of Math | Cool Examples of Quotient Groups | Abstract Algebra @WrathofMath | Uploaded 1 month ago | Updated 2 hours ago
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
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Quotient Groups: youtu.be/MKDAZV0XasI
Coset Property we Used in Proof: youtu.be/v9Hi-xNGO2c
Abstract Algebra Course: youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN
Abstract Algebra Exercises: youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz
Get the textbook for this course! amzn.to/3IjoZaO
0:00 Intro
0:10 Example 1
6:27 Example 2
9:59 Example 3
14:21 Conclusion
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Outro music by Ben Watts and is available for channel members.
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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:
youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join
Quotient Groups: youtu.be/MKDAZV0XasI
Coset Property we Used in Proof: youtu.be/v9Hi-xNGO2c
Abstract Algebra Course: youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN
Abstract Algebra Exercises: youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz
Get the textbook for this course! 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: patreon.com/join/wrathofmathlessons
◆ Donate on PayPal: paypal.me/wrathofmath
Outro music by Ben Watts and is available for channel members.
Follow Wrath of Math on...
● Instagram: instagram.com/wrathofmathedu
● TikTok: tiktok.com/@wrathofmathedu
● X: https://x.com/wrathofmathedu
● Facebook: facebook.com/WrathofMath
