Algebra/Groups

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 $$x^{-1}$$.