Quotient Group Construction

Abstract Algebra · Axiom Academy

LESSON Quotient Group Construction Building G/N through coset multiplication and understanding why normality is essential The key insight is that we're treating entire cosets as single elements in our new group. The subgroup N itself becomes the identity element of G/N, and each coset gN is a distinct element of the quotient. To multiply two cosets aN and bN, we take representatives a and b, multiply them in G to get ab, and then form the coset (ab)N. But there's a critical question: does this definition depend on our choice of representatives? This shows that different representatives a' = an₁ and b' = bn₂ must give the same result: a'b' = (an₁)(bn₂) must be in (ab)N. The operation is well-defined if and only if N is normal. Here's why: when we compute a'b' = (an₁)(bn₂), we need this to equal (ab)n for some n ∈ N. This requires n₁b = bn for some n ∈ N, which means bNb⁻¹ ⊆ N for all b ∈ G. The integers modulo n give us the most familiar quotient group. Here G = ℤ (under addition), N = nℤ = ..., -2n, -n, 0, n, 2n, ... , and the quotient consists of n cosets representing remainders when dividing by n. Since ℤ is abelian, every subgroup is normal, so nℤ is automatically normal in ℤ. The quotient captures the "cyclic" nature of arithmetic modulo n. Consider the dihedral group D₄ (symmetries of a square) with 8 elements. Its center Z(D₄) = e, r² consists of the identity and 180° rotation. The quotient D₄/Z(D₄) has order 4.

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