20.123 (2022)

A finite group is called a $D_\pi$-group if any two of its maximal $\pi$-subgroups are conjugate.
$\qquad$ a) Is it true that for any finite $D_\pi$-group $G$ and a $\pi$-Hall subgroup $H$ of $G$, there are elements $x, y, z \in G$ such that $O_\pi(G) = H \cap H^x \cap H^y \cap H^z$?
$\qquad$ b) Suppose that $G$ is a finite $D_\pi$-group in which all simple non-abelian composition factors are sporadic or alternating groups, and let $H$ be a Hall $\pi$-subgroup of $G$. Is it true that $H \cap H^x \cap H^y = O_\pi(G)$ for some $x, y \in G$?
$\qquad$ c) Suppose that $G$ is a finite $D_\pi$-group with trivial soluble radical in which all simple non-abelian composition factors are sporadic groups, and let $H$ be a Hall $\pi$-subgroup of $G$. Is it true that $H \cap H^g = O_\pi(G)$ for some $g \in G$?