AG11 What is a cyclic Group? What is the order of an element?

#abstractalgebra #cyclicgroup generators of cyclic groups, and order of elements in a group are defined and examples given. If a is an element of order m in a group, it is proved that: (1) The cyclic subgroup generated by a consists of exactly all the elements of the form a^k where k is between 1 and m inclusive. (2) If a^s = e, then s is a multiple of the order of a.#grouptheory Subscribe ‪@Shahriari‬ for math videos at the undergraduate level.#grouptheory 00:00 Introduction 00:15 Outline (   • AG09 What is a Group? Definition & Examples  ) 01:03 Prerequisties (   • AG01 What is Abstract Algebra?  ) 01:38 Review: D_8 (   • AG02 Symmetries Of a Square; D_8 dihedral ...  ) 02:37 Review: S_n (   • AG03 Permutations and Bijections form a Gr...      • AG04 What is the Symmetric Group S_n? Cycl...  ) 02:54 Review: Z/nZ and its group of units (   • AG06 Integers Modulo n, (Z/nZ, +), as a Gr...      • AG07 The Group of Units of integers modulo...  ) 03:07 Review: GL(n, p) (   • AG08 General & Special Linear Groups; Size...  ) 03:22 Exponentiation in Groups 04:02 Example: Exponentiation in S_5 05:22 Rules for Exponentiation 06:26 Guiding Question: What is the smallest subgroup containing a given elements? 06:50 Theorem: Subgroup generated by one element is ... 07:26 Example: Non-zero rationals with multiplication 08:15 Definition: Cyclic Groups 08:56 Example: D_8 not cyclic but has cyclic subgroups 09:38 Example: Z/nZ with addition is cyclic 11:07 Example: Group of units of Z/5Z 12:44 Discussion: Additive and Multiplicative notation 13:51 Theorem on Finite Cyclic Groups 14:57 Proof of Theorem (   • GC1 The Division Algorithm for ordinary in...  ) 18:21 Example: S_5 19:55 Definition: Order of an element 21:12 Example: Integers with addition 21:40 Example: D_8 22:58 Example: Z/6Z with addition modulo 6 24:03 Example: GL(2,3) 25:40 Theorem: If a^s = e, then s is a multiple of o(a) 26:22 Proof of Theorem A series of lectures for a full course on undergraduate abstract algebra based on my book: Shahriar Shahriari, Algebra in Action, A Course in Groups, Rings, and Fields, American Mathematical Society, 2017. https://bookstore.ams.org/amstext-27/ An annotated list of Abstract Algebra Videos: https://pomona.box.com/s/06zu90ewikpn... YouTube Playlist:    • Abstract Algebra, an intro via Actions   Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.