1. Introduction
For a prime number p and an algebraic number field F, let
be the p-class tower, more precisely the unramified Hilbert p-class field tower, that is the maximal unramified pro-p extension, of F. The individual stages
and the Galois groups
of the tower
are described by the derived quotients
,with
, of the p-class tower group
. The purpose of this paper is to report on the most up-to-date theoretical view of p-class towers and the state of the art of actual numerical investigations. After a summary of algebraic and arithmetic foundations in §2, four crucial concepts will illuminate recent innovation and progress in a very ostensive way:
• the Artin limit pattern
of the p-class tower
in §3,
• successive approximation and the current status of computational perspectives in §4,
• maximal subgroups of 3-class tower groups with coclass one in §5, and
• the realization of new 3-class tower groups over dihedral fields in §6.
2. Algebraic and Arithmetic Foundations
2.1. Abelian Type Invariants
First, we recall the concepts of abelian type invariants and abelian quotient invariants in the context of finite p-groups and infinite pro-p groups, and we specify our conventions in their notation.
Let
be a prime number. It is well known that a finite abelian group A with order
a power of p possesses a unique representation
(2.1)
as a direct sum with integers
,
for
, and strictly decreasing
.
Definition 2.1 The abelian type invariants of
are given either in power form,
(2.2)
or in logarithmic form with formal exponents indicating iteration,
(2.3)
Let G be a pro-p group with commutator subgroup
and finite abelianization
.
Definition 2.2 The abelian quotient invariants of G are the abelian type invariants of the biggest abelian quotient of G
(2.4)
2.1.1. Higher Abelian Quotient Invariants of a Pro-p Group
Within the frame of group theory, abelian quotient invariants of higher order are defined recursively in the following manner.
Definition 2.3 The set of all maximal subgroups of G which contain the commutator subgroup,
(2.5)
is called the first layer of subgroups of G. For any positive integer
, abelian quotient invariants of nth order of G are defined recursively by
(2.6)
2.1.2. Higher Abelian Type Invariants of a Number Field
Within the frame of algebraic number theory, abelian type invariants of higher order are defined recursively in the following way.
Let F be an algebraic number field, denote by
the p-class group of F, and by
the first Hilbert p-class field of F, that is, the maximal abelian unramified p-extension of F.
Definition 2.4 The set of all unramified cyclic extensions
of degree p which are contained in the p-class field,
(2.7)
is called the first layer of extension fields of F. For any positive integer
, abelian type invariants of nth order of F are defined recursively by
(2.8)
2.2. Transfer Kernel Type
Next, we explain the concept of transfer kernel type of finite p-groups and infinite pro-p groups.
2.2.1. Transfer Kernel Type of a Pro-p Group
Denote by
a prime number. Let G be a pro-p group with commutator subgroup
and finite abelianization
.
Definition 2.5 By the transfer kernel type of
, we understand the finite family of kernels,
(2.9)
where
denotes the transfer homomorphism from G to the normal subgroup S of finite index
, as given in Formula (3.1).
More specifically, suppose that
is elementary abelian of rank two. Then
has
elements
, the transfer kernel type of G is described briefly by a family of non-negative integers
such that
(2.10)
and the symmetric group
of degree
acts on
via
, for each
, where the extension
of
to
fixes the zero.
Definition 2.6 The orbit
is called the invariant type of G, but it is actually given by one of the orbit representatives
. Any two distinct orbit representatives
are called equivalent, denoted by the symbol
.
2.2.2. Transfer Kernel Type of a Number Field
Let F be an algebraic number field, and denote by
the p-class group of F.
Definition 2.7 By the transfer kernel type of F, we understand the finite family of kernels,
(2.11)
where
denotes the transfer of p-classes from F to the unramified cyclic extension E of degree
, which is also known as the p-class extension homomorphism.
More specifically, suppose that
is elementary abelian of rank two. Then
has
elements
, the transfer kernel type of F is described briefly by a family of non-negative integers
such that
(2.12)
and the symmetric group
of degree
acts on
via
, for each
, where the extension
of
to
fixes the zero.
Definition 2.8 The orbit
is called the invariant type of F, but it is
actually given by one of the orbit representatives
. Any two distinct orbit representatives
are called equivalent, denoted by the
symbol
.
3. The Artin Limit Pattern
Let p be a prime number. For the recursive construction of the Artin limit pattern of a pro-p group G with commutator subgroup
and finite abelianization
, we need the following considerations.
3.1. Mappings of the Artin Limit Pattern
Due to our assumptions, the first layer
of subgroups of G is a finite set consisting of maximal normal subgroups S of G with abelian quotients
. Consequently, the Artin transfer homomorphism from G to
is distinguished by a very simple mapping law:
(3.1)
where h denotes an arbitrary element in
( [1] , 4.1, p. 76).
The Artin limit pattern encapsulates particular group theoretic information (connected with Artin transfers) about the lattice of subgroups of G, where each element U has at least one predecessor, except the root G itself. We select a unique predecessor in the following way: for
we put
, and we add the formal definition
. This enables a recursive construction, as follows:
Definition 3.1 The collection of Artin transfers up to order n of G is defined recursively by
(3.2)
The limit of this infinite recursive nesting process is denoted by
(3.3)
and is called the Artin transfer collection of G.
Remark 3.1 By means of the collection of Artin transfers up to order three,
it should be emphasized that our definition of stepwise relative mappings
and
admits finer information than the corresponding absolute mappings
( [1] , Thm. 3.3, p. 72), since in general the kernel of
cannot be reconstructed from
and
.
3.2. Objects of the Artin Limit Pattern
The infinite collection of mappings
is only the foundation for the objects
and
we are really interested in.
Definition 3.2 The iterated abelian quotient invariants up to order n of G are defined recursively by
(3.4)
Similarly, the iterated transfer kernels up to order n of G are defined recursively by
(3.5)
Both are collected in the nth order Artin pattern
of G. The limits of these infinite recursive nesting processes are called the abelian invariant collection of G,
(3.6)
and the transfer kernel collection of G,
(3.7)
Finally, the pair
is called the Artin limit pattern of G.
Remark 3.2 For a finite p-group G, the recursive nesting processes in the definition of the Artin limit pattern are actually finite.
The abelian quotient invariants are a unary concept, since
depends on G only. The first order abelian quotient invariants
already contain non-trivial information on the abelianization of G.
The transfer kernels are a binary concept for
, since
depends on
and S. The first order transfer kernel of G is trivial:
, and non-trivial information starts with the transfer kernels of second order
for
which are members of
.
The analogous constructions for a number field F instead of a pro-p group G, along the lines of §§2.1.2 and 2.2.2, lead to the Artin limit pattern
of F.
3.3. Connection between Pro-p Groups and Number Fields
Let
be the Hilbert p-class tower of the number field F, that is, the maximal unramified pro-p extension of F, and denote by
its Galois group, which is briefly called the p-tower group of F. Now we are going to employ the abelian type invariant collection
of F, and the abelian quotient invariant collection
of
, i.e., the first component of the respective Artin limit pattern. The transfer kernel collections
will be considered further in §5.
Theorem 3.1 For each integer
, the abelian quotient invariants of nth order of the p-tower group
of F are equal to the abelian type invariants of nth order of the number field F
(3.8)
The invariant type of the p-tower group
of F coincides with the invariant type of the number field F
(3.9)
Even the orbit representatives of the transfer kernel types of
and F coincide,
(3.10)
provided that the
and the
are connected by
, for each
. Otherwise, we only have equivalence
.
Proof. The claims are well-known consequences of the Artin reciprocity law of class field theory [2] [3] .
In contrast to the full p-tower group
, the Galois groups
of the finite stages
of the p-class tower of F, that is, of the higher Hilbert p-class fields of the number field F, in general fail to reveal the abelian type invariants of nth order of the number field F. More precisely, there is a strict upper bound on the order n of the ATI of F which coincide with the AQI of order n of the mth p-class group
of F with a fixed integer
, namely the bound
.
Theorem 3.2 (Successive Approximation Theorem.)
Let F be a number field, p a prime, and m,n integers. The abelian invariant collection
of F is approximated successively by the iterated AQI of sufficiently high p-class groups of F:
(3.11)
However, the transfer kernel type is a phenomenon of second order:
(3.12)
in particular, the metabelian second p-class group
of F is sufficient for determining the transfer kernel type of F.
Proof. This is one of the main results in ( [4] , Thm. 1.19, p. 78) and ( [5] , p. 13).
In general, the upper bound on the order n of the ATI of F in Theorem 3.2 seems to be sharp, in the following sense, where
.
Conjecture 3.1 (Stage Separation Criterion.)
Denote by
the length of the p-class tower of F, that is the derived length
of the p-tower group of F. It is determined in terms of iterated AQI of higher p-class groups of F by the following condition:
(3.13)
The sufficiency of the condition in Conjecture 3.1 is a proven theorem ( [5] , p. 13).
4. Successive Approximation of the p-Class Tower
4.1. Computational Perspectives
Our first attempt to find sound asymptotic tendencies in the distribution of higher non-abelian p-class groups
, with
, among the finite p-groups was planned in 1991 already ( [6] , 3, Remark, p. 77). However, the insurmountable obstacles in the required computations limited the progress for twenty years. In 2012, we finally succeeded in the significant break-through of computing the second 3-class groups
, that is, the metabelianizations
of the 3-class tower groups
of all 4596 quadratic fields
with fundamental discriminants in the remarkable range
and elementary bicyclic 3-class group
of rank two ( [7] , 6, pp. 495-499). The underlying computational techniques were based on the principalization algorithm via class group structure which we had invented in 2009 and implemented by means of the number theoretic computer algebra system PARI/GP [8] in 2010, as described in ( [9] , 5-6, pp. 446-455).
Throughout this paper, isomorphism classes of finite groups G are characterized uniquely by their identifier in the SmallGroups Database [10] [11] , which is denoted by a pair
consisting of the order
and a positive integer i, delimited with angle brackets. The counter
is unique for a fixed value of the order o. In the computational algebra system MAGMA [12] [13] [14] , the upper bound
can be obtained as return value of the function NumberOfSmallGroups(o), provided that IsInSmallGroupDatabase(o) returns true. The identifier of a given finite group G can be retrieved as return value of the function IdentifyGroup(G), provided that Can IdentifyGroup(o) returns true.
4.2. Trivial Towers with
For the decision if the p-class tower of a number field F is trivial with length
it suffices to compute the class number
of the field.
Theorem 4.1 (Trivial p-class tower.)
The p-class tower of a number field F is trivial,
, with length
, if and only if the class number
is not divisible by p, i. e., the p-class number is
.
Proof. The proof consists of a sequence of equivalent statements: The class number satisfies
. Û The p-valuation of
is
. Û The p-class number is
. Û The p-class group
is trivial. Û The p-class rank
is equal
to zero. Û The number of unramified cyclic extensions
of degree p is
. Û The maximal unramified p-extension
of F
coincides with F. Û The Galois group
is trivial. Û The length of the p-class tower is
.
Already C. F. Gauss was able to compute class numbers
of quadratic fields
, at a time when the concept of class field theory was not yet coined. Nowadays, there exist extensive tables of quadratic class numbers which even contain the structures of the associated class groups
. In 1998, Jacobson [15] covered all real quadratic fields with positive discriminants in the range
, and in 2016, Mosunov and Jacobson [16] investigated all imaginary quadratic fields with negative discriminants
. Now we apply these results to class field theory.
Corollary 4.1 (Statistics for
.) The asymptotic proportion of imaginary quadratic fields
, with negative discriminants
, whose class number
is, respectively is not, divisible by
is given as 43.99%, respectively 56.01%, by the heuristics of Cohen, Lenstra and Martinet. In Table 1, the approximations of these theoretical limits by relative frequencies in various ranges
are shown.
Proof. The heuristic asymptotic limits are given in ( [17] , 2, (1.1.c), p. 126). Their approximation by discriminants
with
in ( [18] , Example, p. 843) and ( [6] , 2, Remark, and 3, Remark, p. 77), where
, is still rather far away from the limits. In contrast, the approximations associated with the bounds
and
in ( [16] , p. 2001) are very close already.
4.3. Abelian Single-Stage Towers with
The first stage
of the p-class tower of a number field F is determined by the structure of the p-class group
of F as a finite abelian p-group. This is exactly the first order Artin pattern
(4.1)
since the trivial
does not contain information. However, only in the case of p-class rank one,
, it is warranted that the exact length of the tower is
. A statistical example ( [6] , 2, Remark, p. 77) is shown in Table 2.
Theorem 4.2 A number field F with non-trivial cyclic p-class group
has an abelian p-class tower of exact length
, in fact, the Galois group
is cyclic.
Proof. Suppose that
is non-trivial and cyclic. If the p-class tower had a length
, the second p-class group
would be a
Table 1. Imaginary quadratic fields F with non-trivial, resp. trivial, 3-class tower.
Table 2. Imaginary quadratic fields F with cyclic 3-class tower for
.
non-abelian finite p-group with cyclic abelianization
. However, it is well known that a nilpotent group with cyclic abelianization is abelian, which contradicts the assumption of a length
.
Remark 4.1 We interpret the computation of abelian type invariants
of the Sylow 3-subgroup
of the ideal class group
of a quadratic field
as the determination of the single-stage approximation
of the 3-class tower group
of F. This step yields complete information about the lattice of all unramified abelian 3-extensions
within the Hilbert 3-class field
of F.
4.4. Metabelian Two-Stage Towers with
According to the Successive Approximation Theorem 3.2, the second stage
of the p-class tower of a number field F is determined by the second order Artin pattern
(4.2)
The determination of
for a quadratic field F with 3-class rank
requires the computation of four 3-class groups
of unramified cyclic cubic extensions
and of four transfer kernels
.
Whereas Mosunov and Jacobson [16] were able to determine the class groups
of more than 300 billion, precisely 303963550712, imaginary quadratic fields F with discriminants
by parallel processes on multiple cores of a supercomputer in several years of total CPU time, it is currently definitely out of scope to compute the class groups
, for the 22757307168 unramified cyclic cubic extensions
, of absolute degree six, of the 5689326792 imaginary quadratic fields F with discriminants
and 3-class rank
.
Therefore, it must not be underestimated that Boston, Bush and Hajir [19] succeeded in completing this task for the smaller range
with 461925 imaginary quadratic fields F having 3-class rank
, and 1847700 associated totally complex dihedral fields
of degree six ( [7] , Prp. 4.1, p. 482). For this purpose the authors used the computational algebra system MAGMA [12] [13] [14] in a distributed process involving several processors with multiple cores. 276375 of these quadratic fields F have a 3-class group
.
Imaginary quadratic fields
with negative discriminants
are the simplest number fields with respect to their unit group
, which is a finite torsion group of Dirichlet unit rank zero. This fact has considerable consequences for their p-class tower groups, according to the Shafarevich theorem [20] , corrected in ( [21] , Thm. 5.1, p. 28), [22] .
Theorem 4.3 Among the finite 3-groups G with elementary bicyclic abelianization
of rank two, there exist only two metabelian groups with GI-action (generator inverting action). and relation rank
(so-called Schur s-groups [23] [19] ), namely
and
.
1) These are the groups of smallest order which are admissible as 3-class tower groups
of imaginary quadratic fields F with 3-class group
.
2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.
(a) If
then
.
(b) If
then
.
3) The actual distribution of these 3-class tower groups G among the 276375 imaginary quadratic fields
with 3-class group
and discriminants
is presented in Table 3.
Proof. All finite 3-groups G with abelianization
are vertices of the descendant tree
with abelian root
. A search for metabelian vertices with relation rank
in this tree yields three hits
,
, and
, but only the latter two of them possess a GI-action.
The abelianization
of a finite 3-group G which is realized as the 3-class tower group
of an algebraic number field F is isomorphic to the 3-class group
of F. When F is imaginary quadratic, it possesses signature
and torsionfree Dirichlet unit rank
. If
, then the generator rank of G is
and the Shafarevich theorem implies bounds for the relation rank
.
The entries of Table 3 have been taken from [19] .
More recently, Boston, Bush and Hajir [24] used MAGMA [14] for computing the class groups of the 481756 real quadratic fields F having 3-class rank
and discriminants in the range
, and the class groups of the 1927024 associated totally real dihedral fields
of degree six, arising from unramified cyclic cubic extensions
( [7] , Prp. 4.1, p. 482). 415698 of these quadratic fields F have a 3-class group
(415699 according to( [15] , Tbl. 7)).
Real quadratic fields
with positive discriminants
are the second simplest number fields with respect to their unit group
, which is an infinite group of torsionfree Dirichlet unit rank one. Again, there are remarkable consequences for their p-tower groups, by the Shafarevich theorem ( [21] , Thm. 5.1, p. 28).
Theorem 4.4 Among the finite 3-groups G with elementary bicyclic abelianization
of rank two, there exist infinitely many
Table 3. Frequencies of metabelian 3-class tower groups G for
.
metabelian groups with RI-action and relation rank
(so-called Schur + 1 s-groups [24] ), but only three of minimal order 34, namely
,
and
.
1) These are the groups of smallest order which are admissible as 3-class tower groups
of real quadratic fields F with 3-class group
.
2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.
(a) If
then
.
(b) If
then
.
(c) If
then
.
3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields
with 3-class group
and discriminants
is presented in Table 4. Additionally, the frequencies of the groups
and
in Theorem 4.3 are given.
Proof. A search for metabelian vertices G of minimal order with relation rank
in the descendant tree
with abelian root
yields three hits
,
, and
. All of them possess a RI-action.
The abelianization
of a finite 3-group G which is realized as the 3-class tower group
of an algebraic number field F is isomorphic to the 3-class group
of F. When F is real quadratic, it possesses signature
and torsionfree Dirichlet unit rank
. If
, then the generator rank of G is
and the Shafarevich theorem implies bounds for the relation rank
.
The entries of Table 4 have been taken from [24] .
In [24] , Boston, Bush and Hajir only computed the first component of the second order Artin pattern
in Formula (4.2), that is, the abelian type invariants
of second order of real quadratic fields F with discriminants
. Determining the second component
, the transfer kernel type of F, is considerably harder with respect to the computational expense. Consequently, the most extensive numerical results on transfer kernels available currently, have been computed by ourselves for the smaller ranges
in [25] [26] , and, even computing third order Artin
Table 4. Frequencies of metabelian 3-class tower groups G for
.
patterns, for
in [27] [28] . With the aid of these results, we now illustrate that the transfer kernels
of 3-class extensions
from real quadratic fields F to unramified cyclic cubic extensions
are capable of narrowing down the number of contestants for the 3-tower group
significantly, and thus of refining the statistics in [24] .
Corollary 4.2
1) If
then
.
2) If
then
with
.
3) If
then
with
.
The actual distribution of these 3-class tower groups G among the 34631, respectively 2576, real quadratic fields
with 3-class group
and discriminants
, respectively
, is presented in Table 5.
4.5. Non-Metabelian Three-Stage Towers with
According to the Successive Approximation Theorem 3.2, the third stage
of the p-class tower of a number field F is usually determined by the third order Artin pattern
(4.3)
It is interesting, however, that there are extensive collections of quadratic fields F with 3-class towers of exact length
, which can be characterized by the second order Artin pattern already. We begin with imaginary quadratic fields
with discriminants
.
Theorem 4.5 Among the finite 3-groups G with elementary bicyclic abelianization
of rank two, there exist infinitely many non-
Table 5. Frequencies of metabelian 3-class tower groups G for
, resp. 107.
metabelian groups with GI-action and relation rank
(so-called Schur s-groups [19] [23] ), but only seven of minimal order 38, namely
with
.
1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups
of imaginary quadratic fields F with 3-class group
.
2) Exceptionally, for an imaginary quadratic field F, the trailing six of these groups are determined by the second order Artin pattern already.
(a) If
then
.
(b) If
then
with
.
(c) If
then
.
(d) If
then
with
.
3) The actual distribution of these 3-class tower groups G among the 24476 imaginary quadratic fields
with 3-class group
and discriminants
is presented in Table 6.
Proof. By a similar but more extensive search than in the proof of Theorem 4.3. Data for Table 6 has been computed by ourselves in June 2016 using MAGMA [14] .
Remark 4.2 It should be pointed out that items (1) and (2) of Theorem 4.5 are not valid for real quadratic fields, as documented in ( [29] , Thm. 7.8, p. 162, and Thm. 7.12, p. 165).
The group
belongs to the infinite Shafarevich cover of the metabelian group
with respect to imaginary quadratic fields ( [30] , Cor. 6.2, p. 301), [31] . It shares a common second order Artin pattern with all other elements of the Shafarevich cover. Third order Artin patterns must be used for its identification, as shown in ( [29] , Thm. 7.14, p. 168).
Now we turn to real quadratic fields
with discriminants
.
Theorem 4.6 Among the finite 3-groups G with elementary bicyclic abelianization
of rank two, there exist infinitely many non-
Table 6. Frequencies of non-metabelian 3-class tower groups G for
.
metabelian groups with RI-action and relation rank
(so-called Schur + 1 s-groups [24] ), but only nine of minimal order 37, namely
with
.
1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups
of real quadratic fields F with 3- class group
.
2) Exceptionally, for a real quadratic field F, four of these groups are determined by the second order Artin pattern already.
(a) If
then
with
.
(b) If
then
with
3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields
with 3-class group
and discriminants
is presented in Table 7.
Proof. The claims for transfer kernel type c.18,
, are a consequence of ( [21] , Prp. 7.1, p. 32, Thm. 7.1, p. 33, and Rmk. 7.1, p. 35), those for type c.21,
, have been proved in ( [21] , Prp. 8.1, p. 42, Thm. 8.1, p. 44, and Rmk. 8.2, p. 45). A slightly stronger result is the Main Theorem ( [21] , Thm. 2.1, p. 22).
Remark 4.3 The groups
with
are elements of the infinite Shafarevich cover of the metabelian group
with respect to real quadratic fields.
The group
belongs to the infinite Shafarevich cover of the metabelian group
with respect to real quadratic fields.
These five groups share a common second order Artin pattern with all other elements of the relevant Shafarevich cover. Third order Artin patterns must be employed for their identification, as shown in ( [29] , Thm. 7.13, p. 167, and Thm. 7.15, p. 169).
5. Maximal Subgroups of 3-Groups of Coclass One
Let
be the descending lower central series of the group G, defined recursively by
and
for
, in particular,
is the commutator subgroup of G. A finite p-group G is nilpotent with
for some integer
, which is
Table 7. Frequencies of non-metabelian 3-class tower groups G for
.
called the nilpotency class
of G. When G is of order
, for some integer
, the coclass of G is defined by
and
is called the logarithmic order of G.
Finite 3-groups G with coclass
were investigated by N. Blackburn [32] in 1958. All of these CF-groups, which exclusively have cyclic factors
of their descending central series for
, are necessarily metabelian with second derived subgroup
and abelian commutator subgroup
and possess abelianization
, according to Blackburn [33] .
For the statement of Theorem 5.1, we need a precise ordering of the four maximal subgroups
of the group
, which can be generated by two elements
, according to the Burnside basis theorem. For this purpose, we select the generators
such that
(5.1)
and
, provided that G is of nilpotency class
. Here we denote by
(5.2)
the two-step centralizer of
in G.
Parametrized Presentations of Metabelian 3-Groups
The identification of the groups will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn [32] , Miech [34] , and Nebelung [35] :
(5.3)
where
and
are bounded parameters, and the index of nilpotency
is an unbounded parameter.
The following lemma generalizes relations for second and third powers of generators in ( [27] , Lem. 3.1), [28] .
Lemma 5.1 Let
be a finite 3-group with two generators
. Denote by
the main commutator, and by
and
the two iterated commutators. Then the second and third power of the element
, respectively
, are given by
(5.4)
provided that
is central,
, and
.
Proof. We begin by preparing three commutator relations:
(5.5)
Now we prove the power relations by expanding the power expressions by iterated substitution of the commutator relations in Formula (5.5), always observing that
belongs to the centre,
, and
commute:
Theorem 5.1 Let
be a finite 3-group of coclass
and order
with generators
such that
is contained in the two-step centralizer of G, whereas
, given by a polycyclic power commutator presentation with parameters
,
, and index of nilpotency
.
Then three of the four maximal subgroups,
,
, are non-abelian 3-groups of coclass
, as listed in Table 8 in dependence on the parameters
.
The supplementary Table 9 shows the abelian maximal subgroups of the remaining two extra special 3-group of coclass
and order
.
Proof. For an index of nilpotency
, the first maximal subgroup
Table 8. Non-abelian maximal subgroups
of 3-groups G of coclass 1.
Table 9. Abelian maximal subgroups
of extra special 3-groups G.
of G coincides with the two-step centralizer
of G, which is a nearly homocyclic abelian 3-group
of order
, when
. For
, we have
.
We transform all relations of the group
into relations of the remaining three maximal subgroups
of G.
The polycyclic commutator relations
,
for
, and the nilpotency relation
for the group
, with lower central series
for
, can be used immediately for the subgroup
with lower central series
, where
for
, and
.
For the lower central series of
and
, we must employ the main commutator relation
, and
for
. According to the right product rule for commutators, we have
, for
, but
, and in a similar fashion
, for
, but again exceptionally
. For
, the left product rule for commutators shows
, that is, the slight anomaly for the main commutator disappears in the next step. Thus, the lower central series is
for
,
, where generally
for
, and
for
,
for
. In particular,
and
.
The main commutator relation for all three subgroups
of any group
with
is
, that is
, generally, and it remains to determine
.
For this purpose, we come to the power relations of G,
,
, and
for
, supplemented by (5.4):
and
, and we use these relations to determine
in dependence on
. Generally, we have
for
,
for
, and thus uniformly
.
For
, we uniformly have
, and thus
for all three subgroups. For
, we uniformly have
, and thus
for all three subgroups. For
, we have
, but
,
, and thus
for
but
for
, since
.
For
, we have
, but
, and thus
for
but
for
. For
, we have
, but
, and thus
for
but
for
. For
, we have
,
, and thus
for all three subgroups, again observing that
.
The only 3-groups G of coclass
and order
are the two extra special groups
and
. Since
, all their four maximal subgroups,
,
,
,
, are abelian. For
,
is independent of the other generator, and
for
. However, for
,
, we have
,
, and thus
, whereas
.
6. A General Theorem for Arbitrary Base Fields
Suppose that p is a prime, F is an algebraic number field with non-trivial p-class group
, and E is one of the unramified abelian p-extensions of F. We show that, even in this general situation, a finite p-class tower of F exerts a very severe restriction on the p-class tower of E.
Theorem 6.1 Assume that F possesses a p-class tower
of exact length
for some integer
. Then the Galois group
of the p-class tower of E is a subgroup of index
of the p-class tower group
of F and the length of the p-class tower of E is bounded by
.
Proof. According to the assumptions, there exists a tower of field extensions,
where
enforces the coincidence
of the trailing three fields. Since
, the group index of
in
is equal to the field degree
and
is a subgroup of index
of
. The equality
implies the bound
.
We shall apply Theorem 6.1 to the situation where
,
, and E is an
unramified cyclic cubic extension of F, whence
is a maximal subgroup of
.
6.1. Application to Quadratic Base Fields
Proposition 6.1 Let G be a finite 3-group with elementary bicyclic abelianization
. Then the following conditions are equivalent:
1) The transfer kernel type of G is D.10,
.
2) The abelian quotient invariants of the four maximal subgroups
of G are
.
3) The isomorphism types of the four maximal subgroups of G are
and
.
4) The group G is isomorphic to the Schur s-group
with relation rank
.
Proof. We put
and use the presentation [14]
Then we obtain the maximal subgroups
, since
,
, since
,
, since
,
, since
.
Using Lemma 5.1, and comparing to the abstract presentations [14]
and
,
we conclude
, since
,
, since
,
, since
,
, since
.
Theorem 6.2 Let
be a quadratic field with elementary bicyclic 3-class group
. Then the following conditions are equivalent:
1) The transfer kernel type of F is D.10,
.
2) The abelian type invariants of the 3-class groups
of the four unramified cyclic cubic extensions
are
.
3) The second 3-class group
of F has the maximal subgroups
and
.
4) The 3-class tower group
of F is the Schur s-group
with relation rank
.
Proof. The claims follow from Proposition 6.1 by applying the Successive Approximation Theorem 3.2 of first order.
Corollary 6.1 Let F be a quadratic field which satisfies one of the equivalent conditions in Theorem 6.2. Then the length of the 3-class tower of F is
. The four unramified cyclic cubic extensions
are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank
, and possess 3-class towers of length
. More precisely,
and
with relation rank
, but
and
with relation rank
for
.
Proof. This is a consequence of Theorems 6.1 and 6.2, satisfying the Shafarevich theorem.
Proposition 6.2 Let G be a finite 3-group with elementary bicyclic abelianization
. Then the following conditions are equivalent:
1) The transfer kernel type of G is D.5,
.
2) The abelian quotient invariants of the four maximal subgroups
of G are
.
3) The isomorphism types of the four maximal subgroups of G are
and
.
4) The group G is isomorphic to the Schur s-group
with relation rank
.
Proof. We put
and use the presentation [14]
Similarly as in Proposition 6.1, we obtain the maximal subgroups
,
,
, and
.
Using Lemma 5.1, and comparing to the abstract presentations
and
,
we conclude
, since
,
, since
,
, since
,
, since
.
Theorem 6.3 Let
be a quadratic field with elementary bicyclic 3-class group
. Then the following conditions are equivalent:
1) The transfer kernel type of F is D.5,
.
2) The abelian type invariants of the 3-class groups
of the four unramified cyclic cubic extensions
are
.
3) The second 3-class group
of F has the maximal subgroups
and
.
4) The 3-class tower group
of F is the Schur s-group
with relation rank
.
Proof. The claims follow from Proposition 6.2 by applying the Successive Approximation Theorem 3.2 of first order.
Corollary 6.2 Let F be a quadratic field which satisfies one of the equivalent conditions in Theorem 6.3. Then the length of the 3-class tower of F is
. The four unramified cyclic cubic extensions
are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank
, and possess 3-class towers of length
. More precisely,
and
with relation rank
for
, but
and
with relation rank
for
.
Proof. This is a consequence of Theorems 6.1 and 6.3, satisfying the Shafarevich theorem.
6.2. Application to Dihedral Fields
We recall that a dihedral field E of degree 6 is an absolute Galois extension
with group
. It is a cyclic cubic relative extension
of its unique quadratic subfield
, and it contains three isomorphic, conjugate non-Galois cubic subfields
,
,
. The conductor c of
is a nearly squarefree positive integer with special prime factors, and the discriminants satisfy the relations
and
. Here, we shall always be concerned with unramified extensions, characterized by the conductor
, and thus
, a perfect cube, and equal
.
6.2.1. Totally Complex Dihedral Fields
The computational information on 3-tower groups
of imaginary quadratic fields F in Table 3 admits the purely theoretical deduction of impressive statistics for 3-tower groups
of totally complex dihedral fields E in Table 10 by means of the Corollaries 6.1 and 6.2. We use the crucial new insight that the groups
are maximal subgroups of G, because the extensions
are unramified cyclic of degree 3.
6.2.2. Totally Real Dihedral Fields
The computational information on 3-tower groups
of real quadratic fields F in Table 4 admits the purely theoretical deduction of impressive statistics for 3-tower groups
of totally real dihedral fields E in Table 11 by means of Theorem 5.1. Again, we use the innovative result that the groups
are maximal subgroups of G, since the extensions
are unramified cyclic cubic.
Table 10. Frequencies of dihedral 3-class tower groups S for
.
Table 11. Frequencies of dihedral 3-class tower groups S for
.
The first row of Table 11 reveals extensive realizations of the extraspecial group
as 3-tower group of dihedral fields. This is the first time that
occurs as a 3-tower group. It is forbidden for quadratic fields, and it did not occur for cyclic cubic fields and bicyclic biquadratic fields, up to now.
Theorem 6.4 (A new realization as 3-tower group.) The extraspecial 3-group
of coclass 1 and exponent 3 occurs as 3-class tower group
of totally real dihedral fields E of degree 6.
Proof. The group
possesses the relation rank
. According to the Shafarevich Theorem, it is therefore excluded as 3-tower group
of both, imaginary and real quadratic fields F. However, the combination of Theorem 5.1 and Theorem 6.1 proves its occurrence as 3-class tower group
of totally real dihedral fields E of degree 6, as visualized in Table 11.
Theorem 6.5 (3-class tower groups of totally real dihedral fields.) Let
be a real quadratic field with 3-class group
and fundamental discriminant
. Suppose the second order Artin pattern
is given by the abelian type invariants
and the transfer kernel type
. Let
be the three unramified cyclic cubic relative extensions of F with 3-class group
.
Then
is a totally real dihedral extension of degree 6, for each
,
and the connection between the component
of the third order transfer kernel type
and the 3-class tower group
of
is given in the following way:
(6.1)
Proof. This theorem was expressed as a conjecture in [27] [28] , and is now an immediate consequence of Theorems 6.1 and 5.1.
Remark 6.1 Recall that each unramified cyclic cubic relative extension
,
, gives rise to a dihedral absolute extension
of degree 6, that is an
-extension ( [7] , Prp. 4.1, p. 482). For the trailing three fields
,
, in the stable part of
, i.e. with
, we have constructed the unramified cyclic cubic extensions
,
, and determined the Artin pattern
of
, in particular, the transfer kernel type of
in the fields
of absolute degree 18. The dihedral fields
of degree 6 share a common polarization
, the Hilbert 3-class field of F, which is contained in the relative 3-genus field
, whereas the other extensions
with
are non-abelian over F, for each
. Our computational results underpin Theorem 6.5 concerning the infinite family of totally real dihedral fields
for varying real quadratic fields F.
7. Conclusion
Guided by the Successive Approximation Theorem 3.2 in terms of the Artin limit pattern, we have given a most up-to-date survey concerning the finite 3-groups which are populated most densely by 3-class tower groups
of quadratic number fields
in Sections 4.2-4.5. In particular, the discovery of non-metabelian 3-class towers with exact length
, which is currently the maximal proven finite length, in Theorems 4.5 and 4.6, is entirely due to our cooperation with M. R. Bush, initiated by our joint paper [36] . With Theorems 5.1 and 6.1, we have finally presented a new technique for deriving theoretical conclusions on 3-class towers of dihedral fields with degree six from corresponding results for quadratic fields.
Acknowledgements
The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): P 26008-N25. Indebtedness is expressed for valuable suggestions by the referees.