Page 120 - 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. 120
3.5 Nilpotent groups and p -groups
triangular matrices. It is not hard to see that the class of U in this case is exactly n − 1
showing that there are nilpotent groups of arbitrary class. We note here that in the case
n = 3 we call the group U a Heisenberg group over R.
Observe that Ui consists of all upper unitriangular matrices whose first i − 1 super
diagonals are zero. It easily follows that
Ui /Ui +1 R ⊕ R ⊕ · · · ⊕ R .
n −i times
In the case that R = GF(p ) we find U to be a finite p -group of order p n(n−1)/2. On the
other hand, if R = , then U is a finitely generated torsion-free nilpotent group.
Now, let T denote the group of all upper triangular invertible matrices over R. Let
θ : T → (R∗)n be the projection of a matrix to its diagonal. So, this is an epimorphism
whose kernel is precisely equal to U and whose image is an abelian group. It follows that
T is solvable, with the derived length being no more than [log2(n − 1) + 2].
Properties of nilpotent groups
Lemma 3.5.11 If G is a nilpotent group and 1 = N G , then N ∩ Z (G ) = 1.
PROOF. Let i be the smallest natural number s.t. N ∩ Zi (G ) = 1. Then, [N ∩ Zi (G ),G ]
N ∩ Zi −1(G ) = 1, so that N ∩ Zi (G ) N ∩ Z1(G ) = 1 implying equality.
Corollary 3.5.12 A minimal normal subgroup of a nilpotent group is contained in the
center.
Proposition 3.5.13 If A is a maximal normal abelian subgroup of the nilpotent group G ,
then A = CG (A).
PROOF. Clearly A C = CG (A). Suppose that A = C . Then C /A is a nontrivial normal
subgroup of the nilpotent G /A. By Lemma 3.5.11 there is an A = Ax ∈ (C /A) ∩ Z (G /A).
Now 〈x , A〉 is abelian and normal leading to a contradiction.
Theorem 3.5.14 The following conditions are equivalent for a finite group G :
1. G is nilpotent;
2. every subgroup of G is subnormal;
3. Every proper subgroup H of G is properly contained in its normalizer;
4. Every maximal subgroup of G is normal;
triangular matrices. It is not hard to see that the class of U in this case is exactly n − 1
showing that there are nilpotent groups of arbitrary class. We note here that in the case
n = 3 we call the group U a Heisenberg group over R.
Observe that Ui consists of all upper unitriangular matrices whose first i − 1 super
diagonals are zero. It easily follows that
Ui /Ui +1 R ⊕ R ⊕ · · · ⊕ R .
n −i times
In the case that R = GF(p ) we find U to be a finite p -group of order p n(n−1)/2. On the
other hand, if R = , then U is a finitely generated torsion-free nilpotent group.
Now, let T denote the group of all upper triangular invertible matrices over R. Let
θ : T → (R∗)n be the projection of a matrix to its diagonal. So, this is an epimorphism
whose kernel is precisely equal to U and whose image is an abelian group. It follows that
T is solvable, with the derived length being no more than [log2(n − 1) + 2].
Properties of nilpotent groups
Lemma 3.5.11 If G is a nilpotent group and 1 = N G , then N ∩ Z (G ) = 1.
PROOF. Let i be the smallest natural number s.t. N ∩ Zi (G ) = 1. Then, [N ∩ Zi (G ),G ]
N ∩ Zi −1(G ) = 1, so that N ∩ Zi (G ) N ∩ Z1(G ) = 1 implying equality.
Corollary 3.5.12 A minimal normal subgroup of a nilpotent group is contained in the
center.
Proposition 3.5.13 If A is a maximal normal abelian subgroup of the nilpotent group G ,
then A = CG (A).
PROOF. Clearly A C = CG (A). Suppose that A = C . Then C /A is a nontrivial normal
subgroup of the nilpotent G /A. By Lemma 3.5.11 there is an A = Ax ∈ (C /A) ∩ Z (G /A).
Now 〈x , A〉 is abelian and normal leading to a contradiction.
Theorem 3.5.14 The following conditions are equivalent for a finite group G :
1. G is nilpotent;
2. every subgroup of G is subnormal;
3. Every proper subgroup H of G is properly contained in its normalizer;
4. Every maximal subgroup of G is normal;
