Page 131 - 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. 131
mož Moravec: Some Topics in the Theory of Finite Groups 119
Definition 3.5.36 A topological group G is a pro-p group if it is compact and has a basis of
open neighborhoods of the identity consisting of normal subgroups of G of p -power index.
Definition 3.5.37 An inductively ordered set is a partially ordered set I with the property
that for all i , j ∈ I there exists k ∈ I with k > i and k > j . An inverse system of groups
is a family {Gi | i ∈ I } of groups, where I is an inductively ordered set, with surjections
θi j : Gi → G j whenever i > j , satisfying θi j θj k = θi k for all i > j > k .
Definition 3.5.38 Let {Gi | i ∈ I } be an inverse system of groups. The inverse limit of this
system is
proj limGi = (g i ) ∈ Gi | g i θi j = g j for all i > j ,
i ∈I
equipped with the product topology.
If G is a pro-p group and the set of all normal subgroups of G of p -power index,
then = {G /N | N ∈ } forms an inverse system, where the homomorphisms are the
natural ones. We have that G is the inverse limit of . This property in fact characterizes
pro-p groups.
Definition 3.5.39 If a group is an inverse limit of p -groups of coclass r , then it said to be
a pro-p group of coclass r .
It turns out [7] that every infinite pro-p group S of coclass r determines a maximal
coclass tree (S) in (p, r ), namely, the subtree of (p, r ) consisting of all descendants
of S/γi (S), where i is minimal such that S/γi (S) has coclass r and S/γi (S) is not a quotient
of another infinite pro-p group R of coclass r not isomorphic to S.
In 1980, Leedham-Green and Newman posed five conjectures (A–E) about the stu-
ructure of the coclass graph. These are now all theorems [7]. We state them as follows:
Theorem 3.5.40
E Given p and r , there are only finitely many isomorphism types of infinite solvable
pro-p groups of coclass r .
D Given p and r , there are only finitely many isomorphism types of infinite pro-p
groups of coclass r .
C Pro-p groups of finite coclass are solvable.
B For some function g , every finite p -group of coclass r has derived length bounded
by g (p, r ).
A For some function f , every finite p -group of coclass r has a normal subgroup N of
class 2 (1 if p = 2) whose index is bounded by f (p, r ).
Definition 3.5.36 A topological group G is a pro-p group if it is compact and has a basis of
open neighborhoods of the identity consisting of normal subgroups of G of p -power index.
Definition 3.5.37 An inductively ordered set is a partially ordered set I with the property
that for all i , j ∈ I there exists k ∈ I with k > i and k > j . An inverse system of groups
is a family {Gi | i ∈ I } of groups, where I is an inductively ordered set, with surjections
θi j : Gi → G j whenever i > j , satisfying θi j θj k = θi k for all i > j > k .
Definition 3.5.38 Let {Gi | i ∈ I } be an inverse system of groups. The inverse limit of this
system is
proj limGi = (g i ) ∈ Gi | g i θi j = g j for all i > j ,
i ∈I
equipped with the product topology.
If G is a pro-p group and the set of all normal subgroups of G of p -power index,
then = {G /N | N ∈ } forms an inverse system, where the homomorphisms are the
natural ones. We have that G is the inverse limit of . This property in fact characterizes
pro-p groups.
Definition 3.5.39 If a group is an inverse limit of p -groups of coclass r , then it said to be
a pro-p group of coclass r .
It turns out [7] that every infinite pro-p group S of coclass r determines a maximal
coclass tree (S) in (p, r ), namely, the subtree of (p, r ) consisting of all descendants
of S/γi (S), where i is minimal such that S/γi (S) has coclass r and S/γi (S) is not a quotient
of another infinite pro-p group R of coclass r not isomorphic to S.
In 1980, Leedham-Green and Newman posed five conjectures (A–E) about the stu-
ructure of the coclass graph. These are now all theorems [7]. We state them as follows:
Theorem 3.5.40
E Given p and r , there are only finitely many isomorphism types of infinite solvable
pro-p groups of coclass r .
D Given p and r , there are only finitely many isomorphism types of infinite pro-p
groups of coclass r .
C Pro-p groups of finite coclass are solvable.
B For some function g , every finite p -group of coclass r has derived length bounded
by g (p, r ).
A For some function f , every finite p -group of coclass r has a normal subgroup N of
class 2 (1 if p = 2) whose index is bounded by f (p, r ).
