Definition – Group. – Math-reference.net

Page contents
1. Definition
2. Order of a group
3. Order of group element
3. Types of group
4. Properties

Definition

Let $A$ be a set and let $*$ be an operation on $A$ . If the operation $*$ behaves according to the following group axioms,

(G1) – $*$ is associative.

(G2) – For every element $a\in{A}$ there exists an element $i\in{A}$
such that $i*a=a$ and $a*i=a$ . We call $i=a$ the neutral element (or the identity element). We often write the identity element as $e_A$ , where $A$ denotes the group under consideration.

(G3) – For every element $a\in{A}=a$ , there exists an element $a^-1=a$ which we call a inverse, such that $a*a^-1 = i$ , where $i$ is the neutral element.

If the operation meets these axioms, then we say that the set $A$ together with the operation $*$ form a group, and we denote this as $\langle{A, *}\rangle$ .

Definition – Order of a group

The order of a group (with a finite number of elements), is equal to the number of elements in the group. We denote the order of a group $G$ by $|G|$ , or $ord(G)$ .

The order of a group where the number of elements is infinite, is said to have an infinite order.

Definition – Order of a group element

Let $G$ be a group and let $g\in G$ . We say that the order of $g$ is equal to the smallest possible integer $i$ such that $g^i = e_G$ (see here for clarification on the exponential notation).

Types of group

None of the above axioms require that the operation $*$ is commutative. However, if it happens that $*$ is commutative, then we refer to the group as a commutative group (or an abelian group).

Let $|G| = n$ , where $n$ is finite. If every element $g_i\in{G}$ is such that $g_1 * g_i = g_{i+1}$ , and $g_1 *$

Also see cyclic groups.

Group properties

Let $G$ be a group, and let $X = \{a_1, a_2, ..., a_n\}$