Homomorphism with intersection of all Sylow p-subgroups as kernel?

Question

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups?

I was trying to prove that the intersection of Sylow subgroups is normal like this but couldn't think of one. I ended up showing it was characteristic instead. I'm curious if there is a easy homomorphism with this kernel though.

Answer 1

Well, all characteristic subgroups are necessarily normal, so that suffices. As Taylor's comment points out, using that Sylow subgroups are conjugate you know that $g\left(\bigcap P_i\right)g^{-1}=\bigcap g P_i g^{-1}$, which is an intersection of Sylow subgroups. Indeed, we necessarily have $\bigcap g P_i g^{-1}= \bigcap P_i$: if $g P_i g^{-1} = g P_j g^{-1}$, then $P_i=P_j$, and given any sylow subgroup $P$ then $Q=g^{-1}P g$ is also a Sylow subgroup and $gQg^{-1}=P$.

You can then construct a homomorphism with the appropriate kernel via the coset map.

Answer 2

In addition to the already given answers: $$\cap_{P \in Syl_p(G)}P= core_G(P)=O_p(G)$$ is the unique largest normal $p$-subgroup of $G$. It plays an important role in the theory of groups that are ($p$-)solvable.