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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=ANG(S).

Let π be a set of primes. A subgroup H of a group G is called a π-Hall subgroup if every prime divisor of |H| belongs to π and |G:H| is not divisible by the elements in π.

According to [1], we say that a group G satisfies Eπ if there exists a π-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 π-Hall subgroup of G then HA is a π-Hall subgroup of A.

The following statement is the main result of the talk:

Theorem. Let π be a set of primes and let A be a normal subgroup of a finite Eπ-group G. Then A possesses a π-Hall subgroup H such that G=ANG(H).

We also provide examples showing that the condition GEπ in Theorem is essential and that the equality G=ANG(H) need not hold for every π-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: pdf

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