Lagrange's Theorem and Subgroup Index in Abstract Algebra

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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

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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