Definition 2.10 A group (G, *, 1) consists of a set G with a binary operation * and an identity element 1 satisfying the following three axioms:
- for each element x of G there exists an element x−1, called the inverse of x, such that
- (commutativity), the group is said to be commutative (or Abelian). Axioms 1 and 2 define a semigroup.
(Z, +, 0), (Zn, +, 0), (, ., 1)
The following definitions generalize Definitions 2.9.
- The order of an element x of a finite group G is the least positive integer t such that
- If the order of x is equal to the number n of elements in G, then x is said to be a generator of G.
- If G has a generator, then G is said to be cyclic.
Property 2.6 The order of an element x of a finite group G divides the number of elements in G..
Proof First observe that if H is a subgroup of G, then an equivalence ...