Page 124 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

3.5 Nilpotent groups and p -groups

Proposition 3.5.21 Let G be a finite group. Then G is nilpotent if and only if G ≤ Frat(G ).

PROOF. If G is nilpotent and M a maximal subgroup of G , then G ≤ M . Conversely, if
G ≤ Frat(G ) then every maximal subgroup of G is normal.

gap> G := SmallGroup(96, 10);;
gap> F := FrattiniSubgroup(G);;
gap> StructureDescription(F);
"C4 x C2"

3.5.2 Finite p -groups
Basic properties

Proposition 3.5.22 Let G be a group of order p m+1.

(a) If G is nilpotent of class c > 1, then G /Zc−1(G ) is not cyclic.
(b) c ≤ m .
(c) If 0 ≤ i ≤ j ≤ m + 1, every subgroup of order p i is contained in some subgroup of

order p j .
(d) G has subgroups of every order dividing p m+1.

PROOF. (a) If G /Zc−1(G ) were cyclic, G /Zc−2(G ) would be abelian, hence Zc−1(G ) = G , a
contradiction.

(b) |G : Zc−1(G )| ≥ p 2 by (a), all upper central factors have order ≥ p .
(c) Let H be a subgrup of order p i . As H is subnormal in G , it is a part of a compo-
sition series 1 = H0 ≤ · · · ≤ Hi = H ≤ · · · ≤ Hm+1 = G by Jordan-Hölder’s theorem. All
composition factors have order p , hence the assertion.
(d) Follows from (c).

Lemma 3.5.23 Let G be an elementary abelian p -group. Then Frat(G ) = 1.

PROOF. Let G = C n and let Mi = {(x1, . . . xi −1, 1, xi +1, . . . , xn ) : xj ∈ Cp } for i = 1, . . . , n . Then
p
n
M i are maximal subgroups of G and i =1 Mi = 1, hence Frat(G ) = 1.

Theorem 3.5.24 (The Burnside Basis Theorem) Let G be a finite p -group. Then Frat(G ) =
γ2(G )G p , where G p = 〈g p | g ∈ G 〉. Also if |G : Frat(G )| = p r , then every set of generators of
G has a subset of r elements which also generates G .