An analogue of the Frattini Argument for Hall subgroups

D. Revin,   E. Vdovin

The following simple statement is frequently used in the finite group theory.

The Frattini Argument. Let \(A\) be a normal subgroup of a finite group \(G\) and let \(S\) be a Sylow \(p\)-subgroup of \(A\) for a prime \(p\). Then \(G=AN_G(S)\).

Let \(\pi\) be a set of primes. A subgroup \(H\) of a 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 group \(G\) satisfies \(\mathscr E_\pi\) if there exists a \(\pi\)-Hall subgroup in \(G\).

It is easy to show that if \(A\) is a normal subgroup of a finite group \(G\) and \(H\) is a \(\pi\)-Hall subgroup of \(G\) then \(H\cap A\) is a \(\pi\)-Hall subgroup of \(A\).

The following statement is the main result of the talk:

Theorem. Let \(\pi\) be a set of primes and let \(A\) be a normal subgroup of a finite \(\mathscr E_\pi\)-group \(G\). Then \(A\) possesses a \(\pi\)-Hall subgroup \(H\) such that \(G=AN_G(H)\).

We also provide examples showing that the condition \(G\in\mathscr E_\pi\) in Theorem is essential and that the equality \(G=AN_G(H)\) need not hold for every \(\pi\)-Hall subgroup \(H\) of \(A\).

References

1. P. Hall, Theorems like Sylow's, Proc. London Math. Soc., 6, N22 (1956), 286–304.

See also the authors' pdf version: