Algebra/Groups

------------------------

Algebra Chapter 25: Group Theory Section 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 $a^{-1}$ such that $a^{-1}\bullet a=e$

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