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The Most Important Theorem in Finite Group Theory

Lagrange's Theorem is often described as the most important theorem in finite group theory. If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. A corollary says that the order of any element of G divides the order of G. The converse of Lagrange's Theorem is false. For example, the alternating group A4, which has order 12, has no subgroup of order 6 even though 6 divides 12. The proof of Lagrange's Theorem is based on properties of cosets (left cosets aH and right cosets Ha). In particular, the collection of all left cosets of a subgroup H partitions the group G into subsets with the same cardinality (number of elements). The number of distinct left cosets is called the index of H in G and equals the order of G divided by the order of H. 🔴 "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J 🔴 Abstract Algebra Exam 1 Review Problems and Solutions:    • Abstract Algebra Exam 1 Review Proble...   🔴 Abstract Algebra Exam 2 Review Problems and Solutions:    • Abstract Algebra Exam 2 Review Proble...   🔴 Abstract Algebra Exam 3 Review Problems and Solutions:    • Abstract Algebra Exam 3 Review Proble...   🔴 Abstract Algebra Final Exam Review Problems and Solutions:    • Abstract Algebra Final Exam Review Pr...   🔴 Abstract Algebra Problems with Solutions (including Proofs):    • Abstract Algebra Problems with Soluti...   🔴 Abstract Algebra Lectures Playlist:    • Abstract Algebra Course, Lecture 1: I...   🔴 Check out my blog at: https://infinityisreallybig.com/ #AbstractAlgebra #GroupTheory #LagrangeTheorem Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.