Verifying A₄ is Normal in S₄

Abstract Algebra · Axiom Academy

EXAMPLE Verifying A₄ is Normal in S₄ Step-by-step verification using conjugation invariance and the sign homomorphism Excellent work! You've successfully verified that A₄ is a normal subgroup of S₄. Here's what we learned: Multiple Methods: We used conjugation invariance and the kernel property. We could also use that index 2 subgroups are always normal. Sign Homomorphism: The map sgn: S₄ → ±1 partitions S₄ into even (A₄) and odd permutations. A₄ = ker(sgn). Conjugation Preserves Parity: Conjugating an even permutation by any permutation (even or odd) always yields an even permutation. Kernel Property: Kernels of group homomorphisms are always normal subgroups. This is one of the most powerful results in group theory! Index 2 Criterion: Any subgroup of index 2 is automatically normal. Since |S₄| = 24 and |A₄| = 12, we have [S₄ : A₄] = 2. This result generalizes: Aₙ is normal in Sₙ for all n ≥ 2. The alternating groups form an important class of simple groups (for n ≥ 5), playing a fundamental role in the classification of finite simple groups!

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