Partitioning Groups

Abstract Algebra · Axiom Academy

Discover how subgroups divide groups into equal-sized pieces, like tiles covering a floor with no gaps or overlaps. Let's start with the group ℤ₁₂ (integers modulo 12 under addition). Below, the yellow tiles show a subgroup H = 0, 3, 6, 9 - multiples of 3. Click on any element not in H to "translate" the subgroup. This creates a coset - what happens when we add that element to each member of H. Let's color all the cosets and see how they fit together. The group ℤ₁₂ breaks into exactly 3 cosets. Let's see how the translation works visually. Each coset is a shifted copy of H. The Foundation of Lagrange's Theorem Every element of G belongs to exactly one coset of H. The cosets are disjoint (no overlap) and their union is the entire group. This is called a partition . Each coset has exactly |H| elements. The bijection g + H ↔ H (by mapping g+h ↔ h) shows that translating preserves size. This uniformity is fundamental! Since cosets partition G with equal sizes: |G| = [G:H] × |H| , where [G:H] is the number of distinct cosets (the index). This immediately implies that |H| divides |G| - Lagrange's Theorem!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.