---- > [!definition] Definition. ([[p-Sylow subgroup]]) > Let $G$ be a finite [[group]] with $|G|=p^{r}m$ where $p$ is [[prime number|prime]], $r \geq 1,$ and $p \not{\vert} m$. In other words, $p^r$ is the highest power of $p$ [[divides|dividing]] $G$. [^1] > A **p-Sylow subgroup** of $G$ is a [[subgroup]] $H$ such that $|H|=p^{r}$. ^425d4e [^1]: These are equivalent because $\begin{align} p^{r} & \text{ is not the highest power of }p \text{ dividing } G \\ \iff & |G|= p^{k}n \text{ for some } n \in \mathbb{N} \text{ and $k>r$}\\ \iff & p^{r}m=p^{k}n\\ \iff & m=p^{k-r}n \\ \iff & m = p(p^{k-r-1}n) \iff p | m. \end{align}$ ---- #### ---- #### 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 > ```