Lagrange's Theorem and Subgroup Index in Abstract Algebra

Support the production of this course by joining Wrath of Math to access all my Abstract Algebra videos plus lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Check out my math fashion brand! https://mathshion.com/ Abstract Algebra course: https://www.youtube.com/playlist?list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN Abstract Algebra exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVQNtNnXeHB_1yquKUY98Xz Get the textbook! https://amzn.to/45IvVgH Business Inquiries: wrathofmathlessons@gmail.com We introduce Lagrange's theorem, showing why it is true and follows from previously proven results about cosets. We also investigate groups of prime order, seeing how Lagrange's theorem informs us about every group of prime order - in particular it tells us that any group of prime order p is cyclic and has all of its non-identity elements as generators. Thus, there is only one group (up to isomorphism) of any given prime order. We conclude with a brief discussion of the index of a subgroup H in a group G, showing that the index of a subgroup is equal to the order of the group divided by the order of the subgroup. #abstractalgebra #grouptheory Intro to Cosets in Group Theory: https://youtu.be/kthl-MhIf4s Proof Cosets Partition the Group: https://youtu.be/hK64I36F5zs Order of Cosets is the Same as Subgroup: https://youtu.be/qXZOw-egq04 ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu 0:00 Intro 1:32 Proving Lagrange's Theorem 5:06 Examples 6:42 Groups of Prime Order are Cyclic 10:07 Index of a Subgroup 11:31 Recap