------------------------ Algebra
Chapter 25: Group Theory
Section 3: Groups
Lagrange's Theorem

25.3: Groups

Definition of a GroupEdit

In standard terms, a group G is a set equipped with a binary operation • such that the following properties hold:

  1. The binary operation is closed. That means, for any two values a and b in G, the combined value a • b is also in G.
  2. The binary operation is associative. For any values a, b, c in G, a • (b • c) = (a • b) • c.
  3. There exists a unique identity element e in G such that for all values a in G, a • e = a = e • a.
  4. There exists a unique inverse element   such that  

If the binary operation is commutative, or b • a = a • b, then the group is said to be Abelian.

Practice ProblemsEdit

Problem 25.1 Let   be a group. Prove that the identity element   is unique. Also prove that every element   has a unique inverse, indicated by  .