The Subgroup Test

Abstract Algebra · Axiom Academy

A systematic method to verify whether a subset of a group forms a subgroup using three simple conditions—or just one! Closure: For all a, b ∈ H , we have ab ∈ H Identity: The identity element e ∈ H Inverses: For all a ∈ H , we have a ⁻¹ ∈ H These three conditions are easier to verify than the full group axioms. Associativity is inherited from the parent group G , so we don't need to check it! Let's verify that H = 0, 3, 6, 9 is a subgroup of ℤ₁₂ (integers mod 12 under addition). This elegant test combines all three conditions into one! Here's why it works: Identity: Pick any a ∈ H . Then aa ⁻¹ = e ∈ H Inverses: We have e ∈ H , so ea ⁻¹ = a ⁻¹ ∈ H Closure: If b ⁻¹ ∈ H , then a(b⁻¹) ⁻¹ = ab ∈ H 4. Non-Example: When the Test Fails Consider H = 0, 3, 6 as a subset of ℤ₁₂. Let's check if it's a subgroup.

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