---- > [!definition] Definition. ([[p-group]]) > A **$p$-group** is a [[group]] whose [[order of a group|order]] is a power of a [[prime number]] $p$. > \ > A $p$-group that is a [[subgroup]] of a [[group]] $G$ is a **p-subgroup** of $G$. > [!proposition] Proposition. ($p$-group lemma) > If a p-group [[group action|acts on]] a set $S$ via $\phi:G \to \text{Perm}(S)$, then $|\text{Fix}(\phi)| \equiv |S| \text{ mod } p.$ ^6b5cca > [!proof] Proof of Proposition. > Suppose $|G|=p^{r}$. By the [[orbit-stabilizer theorem]], the only possible [[orbit]] sizes are $1,p,p^{2},\dots,p^{r}$ since the size of any [[orbit]] needs to [[divides|divide]] $p^{r}$. A fixed point must be in an [[orbit]] of size $1$, and every non-fixed point lies in an [[orbit]] of size $p^{k}$. The total number of non-fixed points is hence divisible by $p$, so $|S|\text{ mod }p$ is [[congruent]] to the number of fixed points. > [!basicexample] > $C_{1}=(e)$, $C_{4}$ ([[cyclic group]] of order $4$), $D_{4}$ ([[dihedral group]] of order 8), and $Q$ ([[quaternion group]]) are all $p$-groups. ---- #### ---- #### 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 > ```