Page 122 - 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. 122
3.5 Nilpotent groups and p -groups
Theorem 3.5.17 Let G be a finite group. For a prime p let Op (G ) be the largest normal
p -subgroup of G . The following groups are then equal to Fit(G ):
(a) The direct product of all Op (G ), where p divides |G |.
(b) The intersection of the centralizers of the chief factors of G .
PROOF. (a) If N G is nilpotent, then N = ×Op (N ). As the group Op (N ) is a characteristic
subgroup of N , it follows that Op (N ) G . Therefore Op (N ) ≤ Op (G ), and thus N ≤ ×Op (G ).
(b) Let 1 = G0 ≤ G1 ≤ · · · ≤ Gn = G be a chief series of G and denote
I = CG (Gi +1/Gi ).
i
Since [Gi +1, I ] ≤ Gi for all i , we get γn+1(I ) = 1, hence I ≤ Fit(G ). Conversely, let F =
Fit(G ). Since G1 is a minimal normal subgroup of G , we get either [G1, F ] = 1 or [G1, F ] =
G1. In the latter case, G1 ≤ γc+1(F ) = 1 for some c , a contradiction. Thus [G1, F ] = 1.
Induction on n shows that F ≤ CG (Gi +1/Gi ) for all i .
gap> G := SmallGroup(96, 10);;
gap> IsNilpotent(G);
false
gap> F := FittingSubgroup(G);;
gap> Order(F);
48
gap> StructureDescription(F);
"C12 x C4"
The Frattini subgroup
The Frattini subgroup Frat(G ) of G is the intersection of all maximal subgroups of G (if
G does not have maximal subgroups, then we define Frat(G ) = G ). Clearly Frat(G ) is
a characteristic subgroup of G . We say that g ∈ G is a nongenerator of G if G = 〈g , X 〉
implies G = 〈X 〉 for every X ⊆ G .
Theorem 3.5.18 Frat(G ) equals the set of nongenerators of G .
PROOF. Let g ∈ Frat(G ), G = 〈g , X 〉, but G = 〈X 〉. There exists M ≤ G which is maximal
subject to 〈X 〉 ≤ M and g ∈/ M . M is a maximal subgroup of G , hence g ∈ M , a contradic-
tion.
Let g be a nongenerator and g ∈/ Frat(G ). Thus g ∈/ M for some maximal subgroup
M . It follows 〈g , M 〉 = G , hence G = M , a contradiction.
Proposition 3.5.19 Let G be a finite group.
Theorem 3.5.17 Let G be a finite group. For a prime p let Op (G ) be the largest normal
p -subgroup of G . The following groups are then equal to Fit(G ):
(a) The direct product of all Op (G ), where p divides |G |.
(b) The intersection of the centralizers of the chief factors of G .
PROOF. (a) If N G is nilpotent, then N = ×Op (N ). As the group Op (N ) is a characteristic
subgroup of N , it follows that Op (N ) G . Therefore Op (N ) ≤ Op (G ), and thus N ≤ ×Op (G ).
(b) Let 1 = G0 ≤ G1 ≤ · · · ≤ Gn = G be a chief series of G and denote
I = CG (Gi +1/Gi ).
i
Since [Gi +1, I ] ≤ Gi for all i , we get γn+1(I ) = 1, hence I ≤ Fit(G ). Conversely, let F =
Fit(G ). Since G1 is a minimal normal subgroup of G , we get either [G1, F ] = 1 or [G1, F ] =
G1. In the latter case, G1 ≤ γc+1(F ) = 1 for some c , a contradiction. Thus [G1, F ] = 1.
Induction on n shows that F ≤ CG (Gi +1/Gi ) for all i .
gap> G := SmallGroup(96, 10);;
gap> IsNilpotent(G);
false
gap> F := FittingSubgroup(G);;
gap> Order(F);
48
gap> StructureDescription(F);
"C12 x C4"
The Frattini subgroup
The Frattini subgroup Frat(G ) of G is the intersection of all maximal subgroups of G (if
G does not have maximal subgroups, then we define Frat(G ) = G ). Clearly Frat(G ) is
a characteristic subgroup of G . We say that g ∈ G is a nongenerator of G if G = 〈g , X 〉
implies G = 〈X 〉 for every X ⊆ G .
Theorem 3.5.18 Frat(G ) equals the set of nongenerators of G .
PROOF. Let g ∈ Frat(G ), G = 〈g , X 〉, but G = 〈X 〉. There exists M ≤ G which is maximal
subject to 〈X 〉 ≤ M and g ∈/ M . M is a maximal subgroup of G , hence g ∈ M , a contradic-
tion.
Let g be a nongenerator and g ∈/ Frat(G ). Thus g ∈/ M for some maximal subgroup
M . It follows 〈g , M 〉 = G , hence G = M , a contradiction.
Proposition 3.5.19 Let G be a finite group.
