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 Gsatisfies 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 H∩A 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 G∈Eπ
in Theorem is essential and that the equality G=ANG(H) need not hold for every π-Hall subgroup H of A.
References
P. Hall, Theorems like Sylow's, Proc. London Math. Soc., 6, N22 (1956), 286–304.