# Algebra/Groups

< Algebra

------------------------ | 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:

- 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. - The binary operation is
*associative.*For any values a, b, c in G, a • (b • c) = (a • b) • c. - There exists a unique
*identity element e*in G such that for all values a in G, a • e = a = e • a. - 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 .