I. Groups: Basic Definitions and Examples

Definition 1: An operation on a set G is a function ∗ : G×G→ G. Definition 2: A group is a set G which is equipped with an operation ∗ and a special element e ∈ G, called the identity, such that (i) the associative law holds: for every x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z; (ii) e ∗ x = x = x ∗ e for all x ∈ G; (iii) for every x ∈ G, there is x′ ∈ G (so-called, inverse) with x ∗ x′ = e = x′ ∗ x. Example: Set Operation “+” Operation “∗” Additional Condition N no no — Z yes no — Q yes no “∗” for Q \ {0} R yes no “∗” for R \ {0} R \Q no no —