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17.43 (2010)

Open

Let $\pi$ be a set of primes. Find all finite simple $D_\pi$-groups (see 3.62) in which
$\qquad$ a) every subgroup is a $D_\pi$-group (H. Wielandt);
$\qquad$ b) every subgroup possessing a Hall $\pi$-subgroup is a $D_\pi$-group.

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

Progress

All finite simple $D_\pi$-groups are known (D. O. Revin, Algebra Logic, 47, no. 3 (2008), 210–227).

Comment of 2013: Alternating and sporadic groups for both parts are found in (N. Ch. Manzaeva, Siberian Electron. Math. Rep., 9 (2012), 294–305 (Russian)).

Comment of 2025: Simple groups of Lie type of rank 1 satisfying condition (a) are known (D. O. Revin, V. D. Shepelev, Siberian Math. J., 65, no. 5 (2024), 1187–1194; V. D. Shepelev, Siberian Math. J., 66, no. 4 (2025), 1049–1062).

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