Subgroups Summary

Abstract Algebra · Axiom Academy

Let's review how subgroups reveal internal structure, and how Lagrange's theorem connects subgroup orders to group orders. Definition: A subset H of a group G that is itself a group under the same operation. Notation: H ≤ G Key Property: Subgroups inherit the group structure from the parent group—they're "groups within groups" Trivial Examples: Every group G has at least two subgroups: e and G itself Geometric Intuition: Subgroups represent symmetries within symmetries—rotations within the full symmetry group of a shape Three Conditions: H ≤ G if (1) H is nonempty, (2) H is closed under the operation, and (3) H contains inverses of all its elements One-Step Test: For finite groups, H ≤ G if H is nonempty and closed under the operation (inverses come for free!) Two-Step Test: H ≤ G if (1) e ∈ H, and (2) for all a, b ∈ H, we have ab⁻¹ ∈ H Practical Use: Checking closure is usually the most work; the other conditions are often obvious Cyclic Group: A group G is cyclic if G = ⟨a⟩ = aⁿ | n ∈ ℤ for some element a. Every element is a power of a single generator Classification: Every cyclic group is isomorphic to either ℤ (infinite) or ℤₙ (finite order n) Counting Generators: ℤₙ has φ(n) generators, where φ is Euler's totient function counting integers coprime to n Subgroup Structure: Every subgroup of a cyclic group is cyclic. For ℤₙ, there is exactly one subgroup of order d for each divisor d of n Left Coset: Given H ≤ G and a ∈ G, the left coset is aH = ah | h ∈ H

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