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.

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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

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