Скачать или смотреть AG19 What is a Group Action?

Definition and Examples of Group Actions. We show that every group action gives a #homomorphism from the group to a group of #permutations #grouptheory #abstractalgebra #groups Subscribe ‪@Shahriari‬ for more undergraduate math videos.
00:00 Introduction
00:30 Definition of a Group (   • AG09 What is a Group? Definition & Examples  )
01:22 Discussion: How do we study groups?
02:02 Intuitive idea of an action
02:37 Prototypical examples of an action
03:03 Definition of an action
05:08 Example: Natural action of S_n on {1, ..., n} (   • AG04 What is the Symmetric Group S_n? Cycl...  )
07:09 Example: Action of S_4 on subsets of size 2 of {1, 2, 3, 4}
09:16 Example: D_8 acts on the corners of a square (   • AG02 Symmetries Of a Square; D_8 dihedral ...  )
10:49 Example: D_8 acts on the diagonals of a square
12:14 Lemma: When a group acts, every group element gives a permutation of the set (   • AG03 Permutations and Bijections form a Gr...  )
13:30 Proof of the Lemma
16:10 Example: GL(2, R) acts on R^2 (   • AG08 General & Special Linear Groups; Size...  )
17:35 Theorem: When G acts on a set, we get a group homomorphism from G to the group of permutations of the set
18:42 Proof of Theorem
22:31 Kernel of an action
23:45 Example: Actions show that S_4 has a subgroup isomorphic to D_8 (   • AG05 S_4 has a subgroup isomorphic to D_8  )
25:48 Example: Actions show that a cyclic group of order 2 is a homomorphic image of D_8
27:16 Previous Example gives a glimpse of quotient groups
28:44 Take Home Message

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 five time winner of Pomona College's Wig teaching award.