Chapter 25: Group Theory
Section 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.
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 .