---- > [!definition] Definition. ([[order of an element in a group]]) > The **order** of an element $a$ in a [[group]] $G$ is the smallest $n \in \mathbb{N}$ for which $a^{n}=e$. If no such $n$ exists, we say $a$ has **infinite order**. We write $|g|=n$ or $|g|=\infty$. > >An element of finite order in a [[group]] called a **torsion element**. > \ > Note that the order of all elements in a finite [[group]] is finite. > [!basicnonexample] Warning. > Keyword *smallest*. In general if $g^{n}=e$ for some finite $n$, then $|g|$ [[divides]] $n$. So $g^{N}=e \iff N \text{ is a multiple of } |g|.$ > Indeed, [[division algorithm|write]] $n=q |g|+r$ for some $0\leq r < |g|$; since $g^{r}=g^{n- |g|q} = g^{n}(g^{|g|})^{-q}=e$ with $r < |g|$, $r=0$. ^nonexample > [!basicproperties] Other Properties. > - $|gh|=|hg|$, for $hg = h \ gh \ h^{-1}$ and [[conjugate#^a1150e|conjugate elements]] in a [[group]] have the same order. ^properties ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```