----- > [!proposition] Proposition. ([[the normalizer lemma]]) > **Part 1.** If $H$ is a [[p-group|p-subgroup]] of $G$, then $[N_{G}(H): H] \equiv [G:H] \text{ mod } p.$ > **Part 2.** Suppose $|G|=p^{r}m$, $p \not{|}m$, and $H \leq G$ with $|H|=p^{i}<p^{r}$. Then $H \lneq N_{G}(H)$, and the [[index of a subgroup|index]] $[N_{G}(H):H]$ is a [[divides|multiple]] of $p$. > - **Corollary:** $H=N_{G}(H)$ is impossible, and $p^{i+1}$ [[divides]] $|N_{G}(H)|.$ Part 1: ![[CleanShot 2023-09-22 at [email protected]|500]] Part 2: ![[CleanShot 2023-09-22 at [email protected]]] > [!proof]- Proof. ([[the normalizer lemma]]) > **Part 1.** Consider the *set* $G / H$, note that $|G / H|=[G : H]$ by definition. The [[group]] $H$ [[group action|acts on]] $G / H$ by *right-multiplication*, via $\begin{align} > H \times G / H & \to G / H \\ > (h,Hx) & \mapsto Hxh. > \end{align}$ > The fixed points of this [[group action]] are the right [[coset]]s $Hx$ in the [[normalizer of a subgroup|normalizer]] of $H$ in $G$, $N_{G}(H)$: $\begin{align} > Hxh = Hx \ \fa h \in H & \iff Hxhx ^{-1}=H \ \fa h \in H \\ > & \iff xh x ^{-1} \in H \ \fa h \in H \\ > & \iff x \in N_{G}(H). > \end{align}$ > By [[p-group#^6b5cca|the p-group proposition]], we have that the number of fixed points of the [[group action]], which we just said equals $[N_{G}(H): H]$, equals $|G / H | \text{ mod }p=[G:H] \text{ mod }p$. This is the result. > > **Part 2.** Since $H \trianglelefteq N_{G}(H)$, we create the [[kernel iff normal subgroup|quotient map]] $q: N_{G}(H) \to N_{G}(H) / H, \ \ \ q: g \mapsto gH.$ > The size of the [[quotient group]] is of course $[N_{G}(H): H]$. By **part 1**, $[N_{G}(H): H] \equiv [G : H] \text{ mod }p$. But $H \leq G$, so by [[Lagrange's Theorem]] $[G:H] \text{ mod }p=\frac{|G|}{|H|} \text{ mod }p=0$. So, we see that $[N_{G}(H):H] \text{ mod }p=0$, which implies $[N_{G}(H):H]$ is a multiple of $p$. In particular, $[N_{G}(H):H] \neq 1$, which mandates that $H$ is a *proper* [[subgroup]] of $N_{G}(H)$. ----- #### ---- #### 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 > ```