On the heritability of the property \(D_\pi\) by subgroups in case \(2\in \pi\)

N. Manzaeva

Let \(\pi\) be a set of primes. A subgroup \(H\) of a finite group \(G\) is called a \(\pi\)-Hall subgroup if every prime divisor of \(|H|\) belongs to \(\pi\) and \(|G:H|\) is not divisible by the elements in \(\pi\). According to [1], we say that a finite group \(G\) satisfies \(D_\pi\) (or \(G\) is a \(D_\pi\)-group), if maximal \(\pi\)-subgroups of \(G\) are all conjugate. Notice that Sylow's theorem implies that maximal \(\pi\)-subgroups in \(D_\pi\)-groups are \(\pi\)-Hall subgroups.

We consider the following problem 17.44(b) from the Kourovka Notebook [2].

Problem. Does an overgroup of a \(\pi\)-Hall subgroup in a \(D_\pi\)-group satisfy \(D_\pi\)?

Using the classification of finite simple groups we obtain an affirmative answer to the problem in case \(2\in \pi\).

Theorem. Let \(\pi\) be a set of primes and let \(2\in \pi\). Suppose a finite group \(G\) satisfies \(D_\pi\) and \(H\) is a \(\pi\)-Hall subgroup of \(G\). Then every subgroup \(M\) of \(G\) with \(H\le M\) satisfies \(D_\pi\).

References

1. P. Hall, Theorems like Sylow's, Proc. London Math. Soc. Ser. 3., 6, N22 (1956), 286–304.
2. V. D. Mazurov (ed.), E. I. Khukhro (ed.), The Kourovka Notebook. Unsolved problems in group theory, 17th ed., Sobolev Inst. Mat., Novosibirsk (2010), 136 p.

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