## 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 thatIf, 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 ...

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