Question

Find the order of all elements of S3

Answer 1

The order of an element a of a group is the smallest positive integer $m$ such that $a^m = e$ (where $e$ denotes the identity element of the group, and $a^m$ denotes the product of $m$ copies of $a$ ). If no such $m$ exists, $a$ is said to have infinite order.

All elements of finite groups have finite order.

The order of a group $G$ is denoted by $ord(G)$ or $|G|$ and the order of an element a by $ord(a)$ or $|a|$ .

The symmetric group $S_3$ has the following multiplication table.

This group has six elements, so $ord(S_3) = 6$ .

By definition, the order of the identity $e$ is $1$ . Each of $s, t,$ and $w$ squares to $e$ , so these group elements have order 2. Completing the enumeration, both $u$ and $v$ have order 3, for $u^2 = v$ and $u^3 = vu = e$ , and $v^2 = u$ and $v^3 = uv = e$.