Mod By A Group: Generalized Modular Arithmetic, from Basic Modular Arithmetic Congruence to 'Normal'

This time I wanted to tackle what it means to Mod by a Group (or rather by a subgroup) and how that can give rise to generalized modular arithmetic. I start from basic modular arithmetic congruence in the integers and used that as a vehicle to build up to the idea of 'Normal' in Abstract algebra The video can be broken up into the following sections.

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00:00 Intro
00:47 What is Modular Arithmetic?
02:43 Groups and Modular Arithmetic
05:31 Mod By a Group (+ an example)
12:23 Generalized Modular Arithmetic And Congruence Relations
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The goal of this video is to provide some intuition for cosets, and the value of a normal subgroups in constructing a more general modular arithmetic where instead of strictly using numbers we can use a bit more abstract algebra to do modular arithmetic on groups. In the process of developing these ideas I'll go through an example of modular arithmetic in the Integers which will provide some intuition for cosets and after defining left and right cosets of a group, we will take a look left equivalence and right equivalence which will give us a method for defining what mod-"ing" by a group means in general. Once we have this definition, I'll work through an example of when modularity does not imply the existence of modular arithmetic to finish up by introducing congruence relations and one (of many equivalent) notions of Normal subgroups that will force the existence of a generalized modular arithmetic.

Corrections: When talking about rotations and how a particular one is not in the subgroup K I write H instead of K as the subgroup. It should be K
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#CHALK #GroupTheory #AbstractAlgebra

[DiegoMathemagician]

Yes, please, talk about fractional modularity!

[CHALKND]

That awkward moment when you trip up speaking and miss it in the edit. I'll leave it though, it's good to acknowledge mistakes. 😅😂😅😂

[Ljk]

How the fuck does this dude not have more subs.