📂Abstract Algebra

Order in Abstract Algebra

Definition¹

Order of a group

group $G$ The number of elements is called the order and is denoted by $|G|$ or $\operatorname{ord}(G)$.

Order of an element

For an element $g$ of the group $G$, the smallest positive integer $n$ that satisfies the following equation is called the order of $g$.

$$ g^{n} = e $$

Here $e \in G$ is the identity element. If such an integer does not exist, $g$ is said to have infinite order. The order of $g$ is denoted by $|g|$ or $\operatorname{ord}(g)$.

Properties

If $|G|$ is finite, $G$ is called a finite group.

• For a group $G$ and two elements $a, b \in G$, we have $|ab| = |ba|$.

• In the case of a cyclic group $\braket{a}$, the following holds. $$ \vert\!\braket{a}\!\vert = |a| $$

• Lagrange’s theorem: The order of a subgroup $H \le G$ divides the order of $G$. $$ |H| \mid |G| $$