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18.31 (2014)

Solved

Let $\pi$ be a set of primes. Is it true that in any $D_\pi$-group $G$ (see 3.62) there are three Hall $\pi$-subgroups whose intersection coincides with $O_\pi(G)$?

Contributor: E. P. Vdovin, D. O. Revin

Progress

No, it is not true, for example, for $G$ being an extension of $L_2(27)$ by a field automorphism of order 3, in which $H$ is an extension of a Borel subgroup of $L_2(27)$ by a field automorphism of order 3 (V. I. Zenkov, Siberian Math. J., 63, no. 4 (2022), 720–722).

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