# Algebra/Groups

 ------------------------ AlgebraChapter 25: Group TheorySection 3: Groups Lagrange's Theorem

25.3: Groups

## Definition of a Group

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 ${\displaystyle a^{-1}}$  such that ${\displaystyle a^{-1}\bullet a=e}$

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

## Practice Problems

Problem 25.1 Let ${\displaystyle G}$  be a group. Prove that the identity element ${\displaystyle e\in G}$  is unique. Also prove that every element ${\displaystyle x\in G}$  has a unique inverse, indicated by ${\displaystyle x^{-1}}$ .