2.2.1 Groups

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:

  1. image
  2. image
  3. for each element x of G there exists an element x−1, called the inverse of x, such that


    If, furthermore,

  4. image (commutativity), the group is said to be commutative (or Abelian). Axioms 1 and 2 define a semigroup.

Examples 2.6

(Z, +, 0), (Zn, +, 0), (image, ., 1)

The following definitions generalize Definitions 2.9.

Definitions 2.11

  1. The order of an element x of a finite group G is the least positive integer t such that


  2. 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.
  3. 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 ...

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