Abstract Algebra, Lec 8B: Cyclic Groups: Empirical Observation of Properties скачать в хорошем качестве

Abstract Algebra, Lec 8B: Cyclic Groups: Empirical Observation of Properties

Learn about cyclic groups by examples. "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J. Check out my blog at: https://infinityisreallybig.com/ Abstract Algebra Playlist:    • Abstract Algebra Course, Lecture 1: Introd...   (0:00) Criterion for a^i = a^j. Example: cyclic group of order 8. If |a| = n, then a^i = a^j iff n divides i - j. (5:16) Corollaries: the order of an element and the order of the cyclic subgroup generated by the element are the same; if |a| = n and a^k = e, then n divides k; |ab| divides |a|*|b| when a and b commute; (11:28) Theorem: If |a| = n, then |a^k| = n/gcd(n,k), arrived at by considering the example with |a| = 8. (16:44) If |a| = n and k and n are relatively prime, then a^k is a generator of the cyclic subgroup generated by "a"; also think about this when |a| = 8 and relate to elements of U(8), the group of units modulo 8 under multiplication. (19:37) Corollaries: order of an element divides the order of the group; if |a| = n, then |a^i| = |a^j| iff gcd(n,i) = gcd(n,j); generators of finite cyclic groups. (22:13) Fundamental Theorem of Cyclic Groups: subgroups of cyclic groups are cyclic; the order of any subgroup divides the order of the group; there is exactly one subgroup for each order that divides the order of the group; mention subgroup lattice; (24:53) Euler phi function: make a table of values, state relationship to the number of generators of a cyclic group of order n, and then plot it on Mathematica with DiscretePlot. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.