AG10 A finite semigroup with cancellation aka an associative Latin Square is a group

In the multiplication table of a group, there is no repeats in any row or any column. This is equivalent to the fact that in groups we can cancel like terms on the right or on the left. Moreover, a finite semigroup (nonempty set with an associative binary operation) with cancellation properties is a group. Follows from the fact that in the definition of the group, we only need a left identity and left inverses. Follows that an associative Latin Square is a Group. Full proofs given. Subscribe ‪@Shahriari‬ for math videos at the college level.#grouptheory #abstractalgebra 00:00 Introduction 00:25 What is a group? (   • AG09 What is a Group? Definition & Examples  ) 00:36 Plan for this lecture 01:16 Discussion: Examples, Patterns, and Proofs 01:31 Multiplication table for D_8 (   • AG02 Symmetries Of a Square; D_8 dihedral ...  ) 03:14 No repeats in any row or any column 03:46 Multiplication table for Z/4Z (   • AG06 Integers Modulo n, (Z/nZ, +), as a Gr...  ) 04:00 Multiplication table for S_3 (   • AG04 What is the Symmetric Group S_n? Cycl...  ) 04:29 No repeats equivalent to cancellation laws 05:02 Lemma: Cancellation Properties for Groups 05:21 Proof of Lemma 06:15 Lemma: Identity & Inverses are unique; inverse of an inverse; inverse of a product 07:32 Proof of Lemma 10:15 Discussion: The identity element 12:24 Theorem: To have a group, left identity and left inverses suffice 13:48 Proof of Theorem 15:56 Latin Squares, Semigroups, Monoids, and Groups 16:46 Theorem: A finite semigroup with cancellation properties is a group 17:03 Corollary: A Latin square with the associative property is a group 17:22 Theorem restated 17:52 Proof of Theorem 21:47 Higman-Neumann Theorem: Can define groups using just one axiom! Next Video:    • AG11 What is a cyclic Group? What is the o...   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.