2-Minute Theorems: Lagrange's Theorem | Group Theory | Abstract Algebra | Lagrange | Dogmathic

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No narration, just animation. Two minutes. Cosets in color. Lagrange's Theorem says the order of a subgroup always divides the order of the group. That sounds like a rule someone made up to be annoying, but it falls out of how cosets work, and cosets are just the subgroup shifted by each element.
This video builds the whole picture from scratch using G = Z₁₂ under addition mod 12. We start with H = {0, 4, 8}, a subgroup of order 3, and form every left coset by hand. 1+H = {1, 5, 9}. 2+H = {2, 6, 10}. 3+H = {3, 7, 11}. Four cosets, each with exactly 3 elements, no overlaps, nothing left behind. The group gets carved into identical pieces, and the arithmetic just falls into place.
Three facts do the heavy lifting. Every coset has the same size as H. No two cosets share an element. Together they cover all of G. Put those together and you get |G| = [G:H] · |H|, which in this case is 12 = 4 × 3. The index [G:H] counts the cosets.
The general statement follows the same logic. If G is finite and H is a subgroup, then |H| divides |G|. That's it. The whole theorem lives in the partition.
No narration, just animation. Two minutes. Cosets in color.
Topics covered: Lagrange's theorem, cosets, left cosets, subgroup, group order, index of a subgroup, partition, finite group, Z mod 12, cyclic group, group theory, abstract algebra, visual proof, math animation, discrete math, modular arithmetic

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Properties and Concepts Used:
Lagrange's Theorem
Left cosets
Subgroup (H ≤ G)
Group order |G| and subgroup order |H|
Index of a subgroup [G:H]
Coset partition (equal size, disjoint, full coverage)
Modular arithmetic (addition mod 12)
Cyclic group Z₁₂
Divisibility (|H| divides |G|)
Finite group
Group operation (addition)
Set notation and element enumeration

#GroupTheory #LagrangesTheorem #AbstractAlgebra #dogmathic #2minutetheorems

Видео 2-Minute Theorems: Lagrange's Theorem | Group Theory | Abstract Algebra | Lagrange | Dogmathic канала Dogmathic