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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