Morphisms, rings, and fields | Group theory episode 6
#monoids #grouptheory In the final video of the series, we talk about morphisms, showing a very interesting connection between the structure inside of a group and the world around it. We finish by briefly talking about rings, fields, and vector spaces. I need to make a small correction: The requirements for vector spaces are not complete. Scalar multiplication must also distribute over vector addition. Please support our channel on Patreon, and get early access to new videos: https://www.patreon.com/user?u=86649007 Don't forget that each video comes with links for further study: [VIS] • Visual Group Theory, Lecture 1.1: What is ... ] Visual Group Theory. I have already mentioned this reference before; I repeat it here because it also contains a good introduction to rings & fields. [JUDSON] http://abstract.ups.edu/ Here you can download a free PDF with many great examples. Good for beginners. [MANIAC] • Chapter 1: Symmetries, Groups and Actions ... The essence of group theory: a good series by Mathemaniac. [BARRUS] • Introduction to Group Isomorphisms • Properties of Isomorphic Groups Introduction to isomorphisms. Clearly explains an isomorphism as a dictionary that helps you navigate between two different groups. 0:00 Introduction 0:38 Morphisms 4:10 The exponential function is an isomorphism 4:51 Commuting diagrams 8:38 The generators are all you need 12:10 Re-discovering the orientation of a square 14:54 Every subgroup gives us a morphism 17:38 Every morphism gives us a subgroup 19:45 Inside vs outside 21:17 Rings & fields 27:34 Vector spaces This video is published under a CC Attribution license ( https://creativecommons.org/licenses/...
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