Elementary Theorems

Abstract Algebra · Axiom Academy

LESSON Elementary Theorems in Group Theory Formal proofs of fundamental properties: uniqueness of identity, uniqueness of inverses, the shoe-sock property, and equation solving—all visualized through permutations. Proof Strategy: We'll use proof by contradiction. Suppose there are two identity elements, e and e'. We'll show they must be equal by using the definition of identity on both elements. Suppose e and e' are both identities in G Since e is an identity: e' · e = e' (e acts as identity) Since e' is an identity: e' · e = e (e' acts as identity) Conclusion: The identity is unique ∎ Proof Strategy: Assume an element a has two inverses, b and c. We'll show that b = c by using associativity and the definition of inverse. Suppose b and c are both inverses of a Consider: b = b · e = b · (a · c) By associativity: b · (a · c) = (b · a) · c Since b is an inverse: (b · a) · c = e · c = c Conclusion: Each element has a unique inverse ∎ Intuition: Like taking off shoes and socks—you must reverse the order! If you put on socks then shoes, you take off shoes then socks. This same principle applies to group elements. We need to show (ab) · (b⁻¹a⁻¹) = e By associativity: a · (b · b⁻¹) · a⁻¹ Since b · b⁻¹ = e: a · e · a⁻¹ = a · a⁻¹ Since (ab) · (b⁻¹a⁻¹) = e, we have (ab)⁻¹ = b⁻¹a⁻¹ ∎ 4. Solving Equations in Groups In groups, we can solve equations like ax = b and xa = b uniquely, but we must be careful about left vs. right multiplication (groups aren't always commutative!).

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