Page 115 - 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. 115
mož Moravec: Some Topics in the Theory of Finite Groups 103
property characterizes finite nilpotent groups. We now exhibit a large class of nilpotent
groups:
Lemma 3.5.1 All finite p -groups are nilpotent.
PROOF. We know that Z (G ) is nontrivial by Proposition 3.2.25. Now use induction on the
order of G to show that G /Z (G ) is nilpotent. From here it easily follows that G is nilpotent
as well.
The following is straightforward to prove:
Lemma 3.5.2 Subgroups, homomorphic images and finite direct products of nilpotent
groups are nilpotent.
We note that nilpotency is not closed under extensions, since S3 is an extension of C3
by C2.
Commutators
The theory of nilpotent groups relies significantly on commutator calculus that we briefly
develop here. A simple commutator of length n of elements x1, . . . , xn ∈G is defined in-
ductively by [x1] = x1 and
[x1, x2, . . . , xn ] = [[x1, . . . , xn−1], xn ].
Lemma 3.5.3 Let x , y , z be elements of a group. Then
1. [x , y ] = [y , x ]−1;
2. [x y , z ] = [x , z ]y [y , z ] and [x , y z ] = [x , z ][x , y ]z ;
3. [x , y −1] = ([x , y ]y −1 )−1 and [x −1, y ] = ([x , y ]x −1 )−1;
4. (the Hall-Witt identity) [x , y −1, z ]y [y , z −1, x ]z [z , x −1, y ]x = 1.
PROOF. Let us only sketch the proof of the Hall-Witt identity. Observe that
[x , y −1, z ]y = x −1y −1x z −1x −1y x y −1z y = u −1v,
where u = z x−1 y x and we obtain v by cyclically permuting x , y , z in the definition of u .
The rest is now immediate.
These identities could also be proved using GAP. For example, in order to prove the
identity [x y , z ] = [x , z ]y [y , z ], it suffices that this holds in the free group generated by
x,y,z:
property characterizes finite nilpotent groups. We now exhibit a large class of nilpotent
groups:
Lemma 3.5.1 All finite p -groups are nilpotent.
PROOF. We know that Z (G ) is nontrivial by Proposition 3.2.25. Now use induction on the
order of G to show that G /Z (G ) is nilpotent. From here it easily follows that G is nilpotent
as well.
The following is straightforward to prove:
Lemma 3.5.2 Subgroups, homomorphic images and finite direct products of nilpotent
groups are nilpotent.
We note that nilpotency is not closed under extensions, since S3 is an extension of C3
by C2.
Commutators
The theory of nilpotent groups relies significantly on commutator calculus that we briefly
develop here. A simple commutator of length n of elements x1, . . . , xn ∈G is defined in-
ductively by [x1] = x1 and
[x1, x2, . . . , xn ] = [[x1, . . . , xn−1], xn ].
Lemma 3.5.3 Let x , y , z be elements of a group. Then
1. [x , y ] = [y , x ]−1;
2. [x y , z ] = [x , z ]y [y , z ] and [x , y z ] = [x , z ][x , y ]z ;
3. [x , y −1] = ([x , y ]y −1 )−1 and [x −1, y ] = ([x , y ]x −1 )−1;
4. (the Hall-Witt identity) [x , y −1, z ]y [y , z −1, x ]z [z , x −1, y ]x = 1.
PROOF. Let us only sketch the proof of the Hall-Witt identity. Observe that
[x , y −1, z ]y = x −1y −1x z −1x −1y x y −1z y = u −1v,
where u = z x−1 y x and we obtain v by cyclically permuting x , y , z in the definition of u .
The rest is now immediate.
These identities could also be proved using GAP. For example, in order to prove the
identity [x y , z ] = [x , z ]y [y , z ], it suffices that this holds in the free group generated by
x,y,z:
