Computing Cosets of ⟨r²⟩ in D₄

Abstract Algebra · Axiom Academy

EXAMPLE Computing Cosets of ⟨r²⟩ in D₄ Learn to partition the dihedral group D₄ into cosets of the subgroup ⟨r²⟩ Symmetries of a square: 4 rotations and 4 reflections D₄ is partitioned into 4 distinct cosets, each containing exactly 2 elements: This verifies Lagrange's Theorem: |D₄| = 8 = 4 cosets × 2 elements = [D₄ : ⟨r²⟩] · |⟨r²⟩| Excellent work! You've successfully computed all cosets of ⟨r²⟩ in D₄. Here's what we learned: Left cosets partition the group: Every element of D₄ belongs to exactly one left coset of ⟨r²⟩ Equal size property: Each coset has exactly |⟨r²⟩| = 2 elements, consistent with Lagrange's Theorem Index of the subgroup: [D₄ : ⟨r²⟩] = 4, meaning there are 4 distinct cosets Computing cosets: For left coset g⟨r²⟩, multiply g by each element of ⟨r²⟩ Right vs. left cosets: In D₄, right cosets can differ from left cosets (⟨r²⟩ is not normal) Verification: |D₄| = 8 = [D₄ : ⟨r²⟩] · |⟨r²⟩| = 4 · 2 ✓ This coset structure is fundamental to understanding quotient groups and normal subgroups. The subgroup ⟨r²⟩ generates a partition that helps us understand the structure of D₄!

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