Abstract Algebra, Lec 9B: Powers of Cyclic Group Generators, Subgroup Lattice, Permutations
If |a| = n, then the cyclic subgroup generated by a^k is the same as the cyclic subgroup generated by a^(gcd(n,k)). "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) Correction of the statement related to the lemma from the end of lecture 9A.
(0:44) Make use of the lemma to prove that, if |a| = n, then the cyclic subgroup generated by a^k is the same as the cyclic subgroup generated by a^(gcd(n,k)).
(6:55) Corollaries (the order of an element divides the order of the group and a^k is a generator of the cyclic subgroup generated by a if gcd(n,k) = 1.
(8:25) Briefly discuss Fundamental Theorem of Cyclic Groups.
(10:16) Make a subgroup lattice for a cyclic group of order 24.
(16:05) Quick overview of group theory facts related to the Euler phi function.
(18:57) Define permutations and permutation groups ("groups of permutations") on a set.
(22:11) The symmetric group Sn on n objects is typically considered to be the collection of all permutations on the set A = {1,2,3,...,n}.
(24:18) Elements of this group are often represented either as "arrays" or "cycles". Show just the array notation in this lecture.
(26:19) The order of Sn is n! ("n factorial").
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