## 2.2 ALGEBRA

### 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:

- for each element
*x* of *G* there exists an element *x*^{−1}, called the *inverse* of *x*, such that
If, furthermore,

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

**Examples 2.6**

(*Z*, +, 0), (*Z*_{n}, +, 0), (, ., 1)

The following definitions generalize Definitions 2.9.

**Definitions 2.11**

- 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 ...