I wanted to prove that the intersection of all Sylow $p$-subgroups of a finite group G is a normal subgroup of $G$.
Can someone enlighten me how is this implication possible:
If an automorphism $\sigma$ of $G$ maps every Sylow $p$-subgroup to a Sylow $p$-subgroup, then the image of the intersection of all Sylow $p$-subgroups under $\sigma$ is itself?
Best Answer
Let $N$ be the intersection of all $p$-Sylow subgroups of $G$. If $P$ is a $p$-Sylow subgroup then $N = \cap_{g\in G}gPg^{-1}$. Thus we have for all $x\in G$ that $$ xNx^{-1} = \bigcap_{g\in G}(xg)P(xg)^{-1} = \bigcap_{g\in G}gPg^{-1} = N, $$ so N is normal.
For the question on $\sigma$ see the accepted answer here:
Intersection of all $p$-Sylow subgroups is normal
So in fact, this works and $N$ is a characteristic subgroup and in particular normal.