TITLE: What is the intersection of all Sylow $p$-subgroup's normalizer?
QUESTION [11 upvotes]: Intersection of all Sylow $p$-subgroups is generally denoted by $O_p(G)$ and it is one of the well studied topics in group theory as there are many theorems related to this.

Let $R$ be intersection of all Sylow $p$-subgroup's normalizer in $G$. It is easy to observe that $R$ is a characteristic subgroup of $G$ containing $O_p(G)$. I wonder the properties of $R$ and its relation with $O_p(G)$. 

If anyone can find or observe something about $R$, I would be thankful.

REPLY [2 votes]: $R_p(G)$ is $p$-solvable, but for every odd prime $p$ there is a finite group $G$ such that $R_p(G)$ is not solvable.

We consider solvability properties of $R_p(G) = \bigcap\{ N_G(P^g) : g \in G \}$ where $P$ is a Sylow $p$-subgroup of $G$. By the previous answer, $R_p(G) = Q \ltimes O_p(G)$ is $p$-closed, so definitely $p$-solvable ($p$-length 1, even).
If $p=2$, then clearly $R_2(G)$ is solvable by Feit–Thompson's odd order theorem. In cases where $|G|\leq 1000$, $R_p(G)$ is always solvable. However, in general this need not be true, since we can take $R_p(G) = G$ by taking any $G$ with a normal Sylow $p$-subgroup. Any such group is $p$-solvable, but need not be solvable. For example $G=A_5 \times C_7$ and $p=7$ works.
A slightly less  trivial example (the smallest pefect example in fact) is $G=A_5 \times \operatorname{GL}(3,2)$ with $p=5$ or $p=7$ and $R_p(G)$ is the coprime direct factor. The next smallest perfect example is $G=\operatorname{SL}(2,5) \ltimes \operatorname{GF}(11)^2$ with $p=11$, and then $O_p(G)=P=Z:=\bigcap\{ C_G(P^g) : g \in G\}$ is a Sylow $p$-subgroup, and $R_p(G)/O_p(G)=\operatorname{SL}(2,5)$ is $11$-solvable (being of order coprime to 11) but not solvable.