Abstract:
We give an example of a series of finite $D_\pi$-groups where for each group $G$ in the series and its $\pi$-Hall subgroup $H$, the inequality $H\cap H^x\cap H^y \neq1$ holds for all $x, y \in G$. Thus a negative answer is obtained both to Problem 7.3 by Vdovin and Revin and its analog—Question 18.31 in The Kourovka Notebook. We also describe the subgroups $\operatorname{Min}_G(H,H,H)$ and $\min_G(H,H,H)$.
Keywords:finite group, $D_\pi$-subgroup, intersection of subgroups.
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