When Cosets Form Groups

Abstract Algebra · Axiom Academy

Discover why coset multiplication only works for special subgroups. Not all subgroups create well-defined coset operations! Meet D₄: Symmetries of a Square Before exploring cosets, let's understand our group. D₄ consists of all symmetries of a square: 4 rotations and 4 reflections. Let's take a subgroup H and see how we can partition D₄ into cosets. Choose a subgroup to explore: When Coset Multiplication Fails Let's try multiplying cosets of H = e, s . Pick two representatives and see what happens! ✅ When Coset Multiplication Works Now try with H = e, r, r², r³ . Watch how the multiplication is well-defined! Normal Subgroups and Quotient Groups A subgroup H of G is normal if gH = Hg for all g ∈ G. Equivalently: gHg⁻¹ = H for all g. When H is normal, we can define (aH)(bH) = (ab)H without ambiguity. The choice of representatives doesn't matter! The set of cosets G/H = gH : g ∈ G forms a group under coset multiplication when H is normal. This is called the quotient group . Quotient groups let us study groups by "collapsing" a normal subgroup to the identity. They're fundamental to understanding group structure! Normal: The rotation subgroup e, r, r², r³ ⊲ D₄, giving quotient D₄/H ≅ ℤ₂ Not Normal: The reflection subgroup e, s is not normal because rs ≠ sr Always Normal: e and D₄ itself are always normal (trivial cases)

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