Page 118 - 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. 118
3.5 Nilpotent groups and p -groups
3. the nilpotency class of G equals the length of the upper central series which also
equals the length of the lower central series.
PROOF. (1). This is true for i = 1. Since Gn−i +1/Gn−i ⊂ Z (G /Gn−i ), we have [Gn−i +1,G ] ⊂
Gn−i . By induction,γi +1(G ) = [γi (G ),G ] [Gn−i +1,G ] Gn−i . The item (2) is another
easy induction and (3) follows.
Lemma 3.5.8 (The three subgroup lemma) Let H , K , L G . If two of the commutator
subgroups [H , K , L], [K , L, H ], [L, H , K ] are contained in a normal subgroup of G , then so
is the third one.
PROOF. By Corollary 3.5.5, [H , K , L] is generated by conjugates of commutators of the
form [h, k −1, l ]. Apply the Hall-Witt identity.
Proposition 3.5.9 Let G be a group and i , j ∈ :
1. [γi (G ), γj (G )] γi +j (G ).
2. γi (γj (G )) γi j (G ).
3. [γi (G ),Zj (G )] Zj −i (G ) if j i .
4. Zi (G /Zj (G )) = Zi +j (G )/Zj (G )
PROOF. (1) Both [γi (G ), γj (G ),G ] and [G , γi (G ), γj (G )] are inductively (on j ) contained
in γi +j +1(G ). By the three subgroup lemma the same holds true for [γj (G ),G , γi (G )] =
[γi (G ), γj +1(G )].
(2) This goes by induction on i : γi +1(γj G ) = [γi (γj (G )), γj (G )] [γi j (G ), γj (G )]
γ(i +1)j (G ).
(3) [γi +1(G ),Zj (G )] = [γi (G ),G ,Zj (G )] [G ,Zj (G ), γi (G )][Zj (G ), γi (G ),G ] Zj −i −1(G )
by induction on i .
(4) Induction on i .
Corollary 3.5.10 For any group G we have that G (i ) γ2i (G ). If G is nilpotent of class c ,
then its derived length is at most log2 c + 1.
PROOF. Apply part (2) of the above proposition to
G (i ) = γ2(· · · (γ2(G )) · · · )
i times
3. the nilpotency class of G equals the length of the upper central series which also
equals the length of the lower central series.
PROOF. (1). This is true for i = 1. Since Gn−i +1/Gn−i ⊂ Z (G /Gn−i ), we have [Gn−i +1,G ] ⊂
Gn−i . By induction,γi +1(G ) = [γi (G ),G ] [Gn−i +1,G ] Gn−i . The item (2) is another
easy induction and (3) follows.
Lemma 3.5.8 (The three subgroup lemma) Let H , K , L G . If two of the commutator
subgroups [H , K , L], [K , L, H ], [L, H , K ] are contained in a normal subgroup of G , then so
is the third one.
PROOF. By Corollary 3.5.5, [H , K , L] is generated by conjugates of commutators of the
form [h, k −1, l ]. Apply the Hall-Witt identity.
Proposition 3.5.9 Let G be a group and i , j ∈ :
1. [γi (G ), γj (G )] γi +j (G ).
2. γi (γj (G )) γi j (G ).
3. [γi (G ),Zj (G )] Zj −i (G ) if j i .
4. Zi (G /Zj (G )) = Zi +j (G )/Zj (G )
PROOF. (1) Both [γi (G ), γj (G ),G ] and [G , γi (G ), γj (G )] are inductively (on j ) contained
in γi +j +1(G ). By the three subgroup lemma the same holds true for [γj (G ),G , γi (G )] =
[γi (G ), γj +1(G )].
(2) This goes by induction on i : γi +1(γj G ) = [γi (γj (G )), γj (G )] [γi j (G ), γj (G )]
γ(i +1)j (G ).
(3) [γi +1(G ),Zj (G )] = [γi (G ),G ,Zj (G )] [G ,Zj (G ), γi (G )][Zj (G ), γi (G ),G ] Zj −i −1(G )
by induction on i .
(4) Induction on i .
Corollary 3.5.10 For any group G we have that G (i ) γ2i (G ). If G is nilpotent of class c ,
then its derived length is at most log2 c + 1.
PROOF. Apply part (2) of the above proposition to
G (i ) = γ2(· · · (γ2(G )) · · · )
i times
