Page 123 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 123
mož Moravec: Some Topics in the Theory of Finite Groups 111
(a) If N G , H ≤ G and N ≤ Frat(H ), then N ≤ Frat(G ).
(b) If K G , then Frat(K ) ≤ Frat(G ).
(c) If N G , then Frat(G /N ) ≥ Frat(G )N /N , with equality if N ≤ Frat(G ).
(d) If A is an abelian normal subgroup of G such that Frat(G ) ∩ A = 1, there exists H ≤ G
such that G = H A and H ∩ A = 1.
PROOF.
(a) If not, then there exists a maximal subgroup M such that N ≤ M . Then G = M N ,
H = (H ∩ M )N , thus H ≤ M , therefore N ≤ M , a contradiction.
(b) Apply (a) with N = Frat(K ) and H = K .
(c) By definition.
(d) Let H be minimal subject to G = H A. Then H ∩ A G . If H ∩ A ≤ Frat(H ), then we
claim that H ∩ A = 1 by (a). Namely, if this were false, there would exist a maximal
subgroup M of H such that H ∩ A ≤ M . Then H = M (H ∩ A) and G = M A, contrary
to the minimality of H .
Theorem 3.5.20 (Gaschütz) Let G be a group.
(a) If Frat(G ) ≤ H ≤ G , where H is finite and H / Frat(G ) is nilpotent, then H is nilpotent.
(b) If G is finite, Frat(G ) is nilpotent.
(c) Define FFrat(G ) by FFrat(G )/ Frat(G ) = Fit(G / Frat(G )). If G is finite, then FFrat(G ) =
Fit(G ).
(d) If G is finite, FFrat(G )/ Frat(G ) is the product of all the abelian minimal normal
subgroups of G / Frat(G ).
PROOF.
(a) Let P be a Sylow subgroup of H , F = Frat(G ), and K = P F ≤ H . K /F is a Sylow
subgroup of H /F , hence K /F is characteristic in H /F . Hence K is normal in G . By
the Frattini argument, G = NG (P)K = NG (P)F = NG (P).
(b) Follows from (a).
(c) Denote H = FFrat(G ). H is nilpotent by (a), thus H ≤ Fit(G ).
(d) Taking quotients, we may assume that Frat(G ) = 1. Write L = Fit(G ). L/ Frat(L) is
abelian, hence L ≤ Frat(L) ≤ Frat(G ) = 1. Thus L is abelian. Let N be the product
of all the abelian minimal normal subgroups of G . Then N ≤ L. There exists H ≤ G
such that G = H N and N ∩ H = 1. H ∩ L is normal in H L = G . Since H ∩ L ∩ N = 1,
it follows that H ∩ L = 1 by the minimality. Then L = L ∩ (H N ) = N .
(a) If N G , H ≤ G and N ≤ Frat(H ), then N ≤ Frat(G ).
(b) If K G , then Frat(K ) ≤ Frat(G ).
(c) If N G , then Frat(G /N ) ≥ Frat(G )N /N , with equality if N ≤ Frat(G ).
(d) If A is an abelian normal subgroup of G such that Frat(G ) ∩ A = 1, there exists H ≤ G
such that G = H A and H ∩ A = 1.
PROOF.
(a) If not, then there exists a maximal subgroup M such that N ≤ M . Then G = M N ,
H = (H ∩ M )N , thus H ≤ M , therefore N ≤ M , a contradiction.
(b) Apply (a) with N = Frat(K ) and H = K .
(c) By definition.
(d) Let H be minimal subject to G = H A. Then H ∩ A G . If H ∩ A ≤ Frat(H ), then we
claim that H ∩ A = 1 by (a). Namely, if this were false, there would exist a maximal
subgroup M of H such that H ∩ A ≤ M . Then H = M (H ∩ A) and G = M A, contrary
to the minimality of H .
Theorem 3.5.20 (Gaschütz) Let G be a group.
(a) If Frat(G ) ≤ H ≤ G , where H is finite and H / Frat(G ) is nilpotent, then H is nilpotent.
(b) If G is finite, Frat(G ) is nilpotent.
(c) Define FFrat(G ) by FFrat(G )/ Frat(G ) = Fit(G / Frat(G )). If G is finite, then FFrat(G ) =
Fit(G ).
(d) If G is finite, FFrat(G )/ Frat(G ) is the product of all the abelian minimal normal
subgroups of G / Frat(G ).
PROOF.
(a) Let P be a Sylow subgroup of H , F = Frat(G ), and K = P F ≤ H . K /F is a Sylow
subgroup of H /F , hence K /F is characteristic in H /F . Hence K is normal in G . By
the Frattini argument, G = NG (P)K = NG (P)F = NG (P).
(b) Follows from (a).
(c) Denote H = FFrat(G ). H is nilpotent by (a), thus H ≤ Fit(G ).
(d) Taking quotients, we may assume that Frat(G ) = 1. Write L = Fit(G ). L/ Frat(L) is
abelian, hence L ≤ Frat(L) ≤ Frat(G ) = 1. Thus L is abelian. Let N be the product
of all the abelian minimal normal subgroups of G . Then N ≤ L. There exists H ≤ G
such that G = H N and N ∩ H = 1. H ∩ L is normal in H L = G . Since H ∩ L ∩ N = 1,
it follows that H ∩ L = 1 by the minimality. Then L = L ∩ (H N ) = N .
