Скачать или смотреть Parallel Worlds: Cosets, Equivalence, and Modular Arithmetic

00:00 Intro
01:53 Last Time
02:55 Equivalence Classes
04:57 Quotient Groups
08:21 Cosets
11:55 Operations on Cosets
17:17 Modular Arithmetic as a Quotient Group
19:27 Wrap-Up and Looking Ahead
21:05 Exercises for the Viewer
24:59 Outro

Parallel Worlds of Math — In this episode of From Zero to Zeta, we explore how equivalence relations lead to equivalence classes, how those classes form cosets, and how modular arithmetic naturally appears as a quotient group. By the end, you’ll see why "Z mod n Z" and "Z n" describe the same structure — and what happens when that structure breaks down in more complicated systems.

What’s inside:

How equivalence classes group elements that behave the same way

How subgroups create cosets that divide a group evenly (the foundation of Lagrange’s Theorem)

Why operations on cosets sometimes fail, especially in non-commutative groups like the Rubik’s Cube group

How modular arithmetic emerges from coset addition and becomes a perfect example of a well-defined quotient group

Try it yourself:

Work out the cosets of the subgroups {0,6}, {0,4,8}, {0,3,6,9}, and {0,2,4,6,8,10} in Z12.

In S4 with H = {e, (12), (14), (24), (124), (142)}, compute (13)(23) and (123)(23). Which cosets do they belong to?

Stay connected:
💬 Discord – Ask questions or get help with the exercises:   / discord  

🌐 Website – Lessons, schedule, and tutoring info: https://fromzerotozeta.com

🎮 Zero2ZetaGaming – Chess and strategy streams   / zero2zetagaming  

If you enjoy the series, remember to like and subscribe — it helps grow the community and keeps From Zero to Zeta going strong!

#grouptheory #education #cosets #quotient #quotientgroup #algebra #abstractalgebra #rubikscube #fromzerotozeta #structure #lesson