The intersection of solvable Hall subgroups in finite groups

E. Vdovin

Let $G$ be a finite group and let $\pi$ be a set of primes. Recall that a subgroup $H$ of $G$ is called a $\pi$-Hall subgroup if all prime divisors of $\vert H\vert$ lie in $\pi$, while $\vert G:H\vert$ is divisible by no prime in $\pi$. By $O_\pi(G)$ we denote the $\pi$-radical of $G$. We obtain the following

Theorem. Let $H$ be a solvable $\pi$-Hall subgroup of a finite group $G$. Then there exist five conjugates of $H$ whose intersection equals $O_\pi(G)$, i.e., there exist $x,y,z,t\in G$ such that $$H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G).$$

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