# p-sylow subgroup

• Nov 5th 2008, 06:05 AM
deniselim17
p-sylow subgroup
Let $o(G)=pq$, where $p>q$ are primes.
Since $p$ divides $o(G)$, $G$ contains an element $b$ of order p.
Let $S=$.
Since $S$ is a $p$-sylow subgroup of $G$, the number of conjugates is $1+kp$ for some $k \geqslant 0$.
But $1+kp=[G:N(S)]$ which divides $o(G)=pq$.
Since $(1+kp,p)=1$, then $1+kp$ divides $q$.
Since $q, then $k=0$ and $S$ is normal in $G$.

I don't understand why S is normal in G. If it is because [G:N(S)]=1, why?