Page 130 - 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. 130
3.5 Nilpotent groups and p -groups
Note that the conjugation relations can be rewritten into commutator ones using the
identity x y = x [x , y ], and that the trivial commutator relations are left out.
The above discussion leads to the following:
Theorem 3.5.34 We have that
f (p n ) ≤ p 1 (n 3 −n ) .
6
PROOF. Let G be as above. The isomorphism class of G is determined by the values of βi ,j
and γi ,j ,k . There are at most p choices for each of these (n 3 − n )/6 elements, so there are
at most p 1 (n 3−n ) isomorphism classes of groups of order p n .
6
3.5.4 Coclass
As we have seen so far, there are many p -groups of given order, too many to classify
them all up to isomorphism. In recent years there has been an idea to clasify p -groups
according to coclass. This has lead to coclass theory which has produced some fasci-
nating results. In this section we will briefly describe some of the main features of the
theory, omitting almost all details. We refer to [7] for proofs and further results.
Let G be a group of order p n . Then its nilpotency class c is strictly smaller of n by
Proposition 3.5.22. The difference n − c is called the coclass of G . Finite p -groups of co-
class 1 are also known as p -groups of maximal class. An example of a p -group of maximal
class is Cp Cp ; its order is p p+1 and the nilpotency class is precisely p (exercise).
Example 3.5.35 Define
Q2n = 〈x , y | y 2n−1 = 1, x 2 = y 2n−2 , y x = y −1〉
to be the generalized quaternion group of order 2n (check that this is indeed its order). The
group Q8 is known as the quaternion group. Similarly, the group
SD2n = 〈x , y | y 2n−1 = 1, x 2 = 1, y x = y 2n−2−1〉
is said to be the semi-dihedral group of order 2n . A classical result of the coclass theory
is that 2-groups of maximal class are precisely dihedral, semi-dihedral, and generalized
quaternion 2-groups.
The goal of coclass theory is to study common properties of finite p -groups of fixed
coclass. To this purpose we study the so-called coclass graph (p, r ) whose vertices cor-
respond to the isomorphism types of p -groups of coclass r . Two vertices G and H are
joined by a directed edge from G to H if and only if G =∼ H /γc (H ), where c is the nilpo-
tency class of H . In order to understand this graph, we need a notion of pro-p -groups:
Note that the conjugation relations can be rewritten into commutator ones using the
identity x y = x [x , y ], and that the trivial commutator relations are left out.
The above discussion leads to the following:
Theorem 3.5.34 We have that
f (p n ) ≤ p 1 (n 3 −n ) .
6
PROOF. Let G be as above. The isomorphism class of G is determined by the values of βi ,j
and γi ,j ,k . There are at most p choices for each of these (n 3 − n )/6 elements, so there are
at most p 1 (n 3−n ) isomorphism classes of groups of order p n .
6
3.5.4 Coclass
As we have seen so far, there are many p -groups of given order, too many to classify
them all up to isomorphism. In recent years there has been an idea to clasify p -groups
according to coclass. This has lead to coclass theory which has produced some fasci-
nating results. In this section we will briefly describe some of the main features of the
theory, omitting almost all details. We refer to [7] for proofs and further results.
Let G be a group of order p n . Then its nilpotency class c is strictly smaller of n by
Proposition 3.5.22. The difference n − c is called the coclass of G . Finite p -groups of co-
class 1 are also known as p -groups of maximal class. An example of a p -group of maximal
class is Cp Cp ; its order is p p+1 and the nilpotency class is precisely p (exercise).
Example 3.5.35 Define
Q2n = 〈x , y | y 2n−1 = 1, x 2 = y 2n−2 , y x = y −1〉
to be the generalized quaternion group of order 2n (check that this is indeed its order). The
group Q8 is known as the quaternion group. Similarly, the group
SD2n = 〈x , y | y 2n−1 = 1, x 2 = 1, y x = y 2n−2−1〉
is said to be the semi-dihedral group of order 2n . A classical result of the coclass theory
is that 2-groups of maximal class are precisely dihedral, semi-dihedral, and generalized
quaternion 2-groups.
The goal of coclass theory is to study common properties of finite p -groups of fixed
coclass. To this purpose we study the so-called coclass graph (p, r ) whose vertices cor-
respond to the isomorphism types of p -groups of coclass r . Two vertices G and H are
joined by a directed edge from G to H if and only if G =∼ H /γc (H ), where c is the nilpo-
tency class of H . In order to understand this graph, we need a notion of pro-p -groups:
