Group Theory
★★★★☆Undergraduate
📖Definition
A group is a set with a binary operation satisfying associativity, existence of identity, and existence of inverses. It's the fundamental structure for studying symmetry mathematically.
📐Formulas
(a · b) · c = a · (b · c)
Associativity
∃ e : a · e = e · a = a
Existence of identity
∀ a, ∃ a⁻¹ : a · a⁻¹ = a⁻¹ · a = e
Existence of inverse
a · b = b · a
Commutativity (for abelian groups)
✏️Examples
예제 1
Show (ℤ, +) is a group.
예제 2
What is the order of S₃ (symmetric group on 3 elements)?
📜History
Discovered by: Évariste Galois (1830s)
Galois introduced group concept while studying polynomial equations. Abel independently developed similar ideas.
⚡Applications
Physics
Symmetry and conservation laws
Cryptography
Elliptic curves, Diffie-Hellman
Chemistry
Molecular symmetry, crystallography
🔗Related Documents
→Prerequisites
←Next Topics
↔Related
#군#대칭#group#symmetry