Making Subgroup Lattices is Fun! (Construct Subgroup Lattice for Cyclic Group ℤ20)

The Fundamental Theorem of Cyclic Groups (FTCG) is very helpful to construct the subgroup lattice for a cyclic group. This is because there is exactly one subgroup of each order that divides the order of the whole group. For the cyclic group of order 20, ℤ20 = ＜1＞={0,1,2,3,...,19} under addition modulo 20, the unique subgroup of order 10 is ＜2＞={0,2,4,6,8,10,12,14,16,18}, the unique subgroup of order 5 is ＜4＞={0,4,8,12,16}, the unique subgroup of order 4 is ＜5＞={0,5,10,15}, the unique subgroup of order 2 is ＜10＞={0,10}, and the unique subgroup of order 1 is the trivial subgroup ＜0＞={0}. We can form a directed graph from these subgroups using set inclusion. This is the subgroup lattice. It's pretty tricky and easy to make a mistake, so be careful!    • Abstract Algebra Course, Lecture 1: Introd...  . 🔴 Joe Gallian's Abstract Algebra textbook: https://amzn.to/4boQZcy 🔴 Abstract Algebra Course Lectures Playlist:    • Abstract (Modern) Algebra Course Lectures   🔴 Number Theory Examples and Proofs Playlist:    • Number Theory Examples and Proofs   🔴 Abstract Algebra Exam 1 Review:    • Abstract Algebra Exam 1 Review Problems an...   ' #AbstractAlgebra #GroupTheory #Subgroup 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.