Question

Question: Let $K$ be a Sylow p-subgroup of $G$and $N$a normal subgroup of $G$. If $K$is a normal subgroup of $N$, prove that $K$ is normal in $G$ .

Short answer

It is proved that,$K$is normal in $G$.

Step-by-step solution

Prove that K is normal in G

Given that K is a sylow p-subgroup of G and N is a normal subgroup of G.

Since $N$is normal, $g^{-1}Ng=N$.

Since $K$is a Sylow p-subgroup of $G$contained in $N$, we have, $\sout{that}$$g^{-1}Kg\underline{\subset }N$.

$\sout{Butn}$Note that $g^{-1}Kg$has the same order as $K$ and therefore, is also a Sylow p-subgroup of role="math" localid="1653323246158" $G$. role="math" localid="1653323333095" $-\sout{andthat}\sout{b}$Both $K$and $g^{-1}Kg$ are also Sylow p-subgroups of $N$.

Since we are assuming $K$is normal in $N$ , then by corollary 9.16, is the only Sylow p-subgroup of $N$and therefore, $g^{-1}Kg=K$.

Since $g\in G$ was arbitrary, we conclude that $K$ is normal in $G$.