_{1}

Let
F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer
n ≥ 1, finitely many candidates for the Galois group
of the
nth stage
of the
p-class tower
over
F are determined by abelian type invariants of
p-class groups C1
_{p}
E of unramified extensions
E/F with degree [
E :
F] =
p
^{n-1}. Illustrated by the most extensive numerical results available currently, the transfer kernels (
T_{E, F}) of the
p-class extensions
T_{E, F} : C1
_{p}
F → C1
_{p}
E from
F to unramified cyclic degree-
p extensions
E/
F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups
S <
G of all 3-groups
G with coclass cc(
G) = 1, and establishing a general theorem on the connection between the
p-class towers of a number field F and of an unramified abelian
p-extension
E/F, we are able to provide a theoretical proof of the realization of certain 3-groups
S with maximal class by 3-tower groups
of dihedral fields
E with degree 6, which could not be realized up to now.

For a prime number p and an algebraic number field F, let F p ( ∞ ) 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 F p ( n ) and the Galois groups Gal ( F p ( n ) / F ) of the tower

F = F p ( 0 ) ≤ F p ( 1 ) ≤ F p ( 2 ) ≤ ⋯ ≤ F p ( n ) ≤ ⋯ ≤ F p (∞)

are described by the derived quotients G / G ( n ) ≃ G p n F : = Gal ( F p ( n ) / F ) ,with n ≥ 1 , of the p-class tower group G : = G p ∞ F : = Gal ( F p ( ∞ ) / F ) . 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 ( τ ( ∞ ) F , ϰ ( ∞ ) F ) of the p-class tower F p ( ∞ ) 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.

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 p ≥ 2 be a prime number. It is well known that a finite abelian group A with order | A | a power of p possesses a unique representation

A ≃ ⊕ i = 1 s ( ℤ / p e i ℤ ) r i (2.1)

as a direct sum with integers s ≥ 0 , r i ≥ 1 for 1 ≤ i ≤ s , and strictly decreasing e 1 > ⋯ > e s ≥ 1 .

Definition 2.1 The abelian type invariants of A are given either in power form,

ATI ( A ) : = [ p e 1 , ⋯ , p e 1 ︷ r 1 times , ⋯ , p e s , ⋯ , p e s ︷ r s times ] , (2.2)

or in logarithmic form with formal exponents indicating iteration,

ATI ( A ) : = [ e 1 r 1 , ⋯ , e s r s ] . (2.3)

Let G be a pro-p group with commutator subgroup G ′ and finite abelianization G a b : = G / G ′ .

Definition 2.2 The abelian quotient invariants of G are the abelian type invariants of the biggest abelian quotient of G

AQI ( G ) : = ATI ( G a b ) . (2.4)

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,

Lyr 1 G : = { S ⊲ G | G ′ ≤ S , ( G : S ) = p } , (2.5)

is called the first layer of subgroups of G. For any positive integer n ≥ 1 , abelian quotient invariants of nth order of G are defined recursively by

τ ( 1 ) G : = AQI ( G ) and τ ( n ) G : = ( τ ( 1 ) G ; ( τ ( n − 1 ) S ) S ∈ Lyr 1 G ) for n ≥ 2. (2.6)

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 Cl p F the p-class group of F, and by F p ( 1 ) 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 E / F of degree p which are contained in the p-class field,

Lyr 1 F : = { E > F | E ≤ F p ( 1 ) , [ E : F ] = p } (2.7)

is called the first layer of extension fields of F. For any positive integer n ≥ 1 , abelian type invariants of nth order of F are defined recursively by

τ ( 1 ) F : = ATI ( Cl p F ) and τ ( n ) F : = ( τ ( 1 ) F ; ( τ ( n − 1 ) E ) E ∈ Lyr 1 F ) for n ≥ 2. (2.8)

Next, we explain the concept of transfer kernel type of finite p-groups and infinite pro-p groups.

Denote by p ≥ 2 a prime number. Let G be a pro-p group with commutator subgroup G ′ and finite abelianization G a b = G / G ′ .

Definition 2.5 By the transfer kernel type of G , we understand the finite family of kernels,

ϰ ( G ) : = ( k e r ( T G , S ) ) S ∈ Lyr 1 G , (2.9)

where T G , S : G / G ′ → S / S ′ denotes the transfer homomorphism from G to the normal subgroup S of finite index ( G : S ) = p , as given in Formula (3.1).

More specifically, suppose that G a b ≃ C p × C p is elementary abelian of rank two. Then Lyr 1 G has p + 1 elements S 1 , ⋯ , S p + 1 , the transfer kernel type of G is described briefly by a family of non-negative integers ϰ ( G ) = ( ϰ i ) 1 ≤ i ≤ p + 1 ∈ [ 0, p + 1 ] p + 1 such that

ϰ i : = ( 0 if k e r ( T G , S i ) = G / G ′ , j if k e r ( T G , S i ) = S j / G ′ for some 1 ≤ j ≤ p + 1, (2.10)

and the symmetric group S p + 1 of degree p + 1 acts on [ 0, p + 1 ] p + 1 via ϰ ↦ ϰ π : = π 0 − 1 ∘ ϰ ∘ π , for each π ∈ S p + 1 , where the extension π 0 of π to [ 0, p + 1 ] fixes the zero.

Definition 2.6 The orbit ϰ ( G ) S p + 1 is called the invariant type of G, but it is actually given by one of the orbit representatives ( ϰ i ) 1 ≤ i ≤ p + 1 . Any two distinct orbit representatives λ 1 , λ 2 ∈ ϰ ( G ) S p + 1 are called equivalent, denoted by the symbol λ 1 ~ λ 2 .

Let F be an algebraic number field, and denote by Cl p F the p-class group of F.

Definition 2.7 By the transfer kernel type of F, we understand the finite family of kernels,

ϰ ( F ) : = ( k e r ( T F , E ) ) E ∈ Lyr 1 F , (2.11)

where T F , E : Cl p F → Cl p E denotes the transfer of p-classes from F to the unramified cyclic extension E of degree [ E : F ] = p , which is also known as the p-class extension homomorphism.

More specifically, suppose that Cl p F ≃ C p × C p is elementary abelian of rank two. Then Lyr 1 F has p + 1 elements E 1 , ⋯ , E p + 1 , the transfer kernel type of F is described briefly by a family of non-negative integers

ϰ ( F ) = ( ϰ i ) 1 ≤ i ≤ p + 1 ∈ [ 0, p + 1 ] p + 1 such that

ϰ i : = ( 0 if k e r ( T F , E i ) = Cl p F , j if k e r ( T F , E i ) = Norm E j / F ( Cl p E j ) for some 1 ≤ j ≤ p + 1, (2.12)

and the symmetric group S p + 1 of degree p + 1 acts on [ 0, p + 1 ] p + 1 via ϰ ↦ ϰ π : = π 0 − 1 ∘ ϰ ∘ π , for each π ∈ S p + 1 , where the extension π 0 of π to [ 0, p + 1 ] fixes the zero.

Definition 2.8 The orbit ϰ ( F ) S p + 1 is called the invariant type of F, but it is

actually given by one of the orbit representatives ( ϰ i ) 1 ≤ i ≤ p + 1 . Any two distinct orbit representatives λ 1 , λ 2 ∈ ϰ ( F ) S p + 1 are called equivalent, denoted by the

symbol λ 1 ~ λ 2 .

Let p be a prime number. For the recursive construction of the Artin limit pattern of a pro-p group G with commutator subgroup G ′ and finite abelianization G a b = G / G ′ , we need the following considerations.

Due to our assumptions, the first layer Lyr 1 G of subgroups of G is a finite set consisting of maximal normal subgroups S of G with abelian quotients G / S . Consequently, the Artin transfer homomorphism from G to S ∈ Lyr 1 G is distinguished by a very simple mapping law:

T G , S : G / G ′ → S / S ′ , g ⋅ G ′ ↦ ( g p ⋅ S ′ if g ∈ ( G / G ′ ) \ ( S / G ′ ) , g 1 + h + h 2 + ⋯ + h p − 1 ⋅ S ′ if g ∈ S / G ′ , (3.1)

where h denotes an arbitrary element in G \ S ( [

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 U ∈ Lyr 1 S we put π ( U ) : = S , and we add the formal definition π ( G ) : = G . 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

α ( 1 ) G : = T π ( G ) , G and α ( n ) G : = [ α ( 1 ) G ; ( α ( n − 1 ) S ) S ∈ Lyr 1 G ] for n ≥ 2. (3.2)

The limit of this infinite recursive nesting process is denoted by

α ( ∞ ) G : = l i m n → ∞ α ( n ) G (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,

α ( 3 ) G = [ T G , G ; ( α ( 2 ) S ) S ∈ Lyr 1 G ] = [ T G , G ; ( [ T G , S ; ( T S , U ) U ∈ Lyr 1 S ] ) S ∈ Lyr 1 G ] ,

it should be emphasized that our definition of stepwise relative mappings T G , S and T S , U admits finer information than the corresponding absolute mappings T G , U = T S , U ∘ T G , S ( [

The infinite collection of mappings α ( ∞ ) G is only the foundation for the objects τ ( ∞ ) G and ϰ ( ∞ ) G we are really interested in.

Definition 3.2 The iterated abelian quotient invariants up to order n of G are defined recursively by

τ ( 1 ) G : = AQI ( G ) and τ ( n ) G : = [ τ ( 1 ) G ; ( τ ( n − 1 ) S ) S ∈ Lyr 1 G ] for n ≥ 2. (3.4)

Similarly, the iterated transfer kernels up to order n of G are defined recursively by

ϰ ( 1 ) G : = k e r ( T π ( G ) , G ) and ϰ ( n ) G : = [ ϰ ( 1 ) G ; ( ϰ ( n − 1 ) S ) S ∈ Lyr 1 G ] for n ≥ 2. (3.5)

Both are collected in the nth order Artin pattern AP ( n ) G : = ( τ ( n ) G , ϰ ( n ) G ) of G. The limits of these infinite recursive nesting processes are called the abelian invariant collection of G,

τ ( ∞ ) G : = l i m n → ∞ τ ( n ) G , (3.6)

and the transfer kernel collection of G,

ϰ ( ∞ ) G : = l i m n → ∞ ϰ ( n ) G . (3.7)

Finally, the pair ALP ( G ) : = ( τ ( ∞ ) G , ϰ ( ∞ ) G ) 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 τ ( 1 ) G = AQI ( G ) = ATI ( G / G ′ ) depends on G only. The first order abelian quotient invariants τ ( 1 ) G already contain non-trivial information on the abelianization of G.

The transfer kernels are a binary concept for S < G , since ϰ ( 1 ) S = k e r ( T π ( S ) , S ) depends on π ( S ) and S. The first order transfer kernel of G is trivial: ϰ ( 1 ) G = k e r ( T π ( G ) , G ) = k e r ( T G , G ) = k e r ( id G / G ′ ) = 1 , and non-trivial information starts with the transfer kernels of second order ϰ ( 1 ) S = k e r ( T π ( S ) , S ) = k e r ( T G , S ) for S ∈ Lyr 1 G which are members of ϰ ( 2 ) G .

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 ALP ( F ) : = ( τ ( ∞ ) F , ϰ ( ∞ ) F ) of F.

Let F p ( ∞ ) be the Hilbert p-class tower of the number field F, that is, the maximal unramified pro-p extension of F, and denote by G p ∞ F = Gal ( F p ( ∞ ) / F ) its Galois group, which is briefly called the p-tower group of F. Now we are going to employ the abelian type invariant collection τ ( ∞ ) F of F, and the abelian quotient invariant collection τ ( ∞ ) ( G p ∞ F ) of G p ∞ F , i.e., the first component of the respective Artin limit pattern. The transfer kernel collections ϰ ( ∞ ) will be considered further in §5.

Theorem 3.1 For each integer n ≥ 1 , the abelian quotient invariants of nth order of the p-tower group G p ∞ F of F are equal to the abelian type invariants of nth order of the number field F

( ∀ n ≥ 1 ) τ ( n ) ( G p ∞ F ) = τ ( n ) F and thus τ ( ∞ ) ( G p ∞ F ) = τ ( ∞ ) F . (3.8)

The invariant type of the p-tower group G p ∞ F of F coincides with the invariant type of the number field F

ϰ ( G p ∞ F ) S p + 1 = ϰ ( F ) S p + 1 . (3.9)

Even the orbit representatives of the transfer kernel types of G p ∞ F and F coincide,

ϰ ( G p ∞ F ) = ( k e r ( T G p ∞ F , U i ) ) 1 ≤ i ≤ p + 1 = ( k e r ( T F , E i ) ) 1 ≤ i ≤ p + 1 = ϰ ( F ) , (3.10)

provided that the U i ∈ Lyr 1 ( G p ∞ F ) and the E i ∈ Lyr 1 F are connected by U i = Gal ( F p ( ∞ ) / E i ) , for each 1 ≤ i ≤ p + 1 . Otherwise, we only have equivalence ϰ ( G p ∞ F ) ~ ϰ ( F ) .

Proof. The claims are well-known consequences of the Artin reciprocity law of class field theory [

In contrast to the full p-tower group G = G p ∞ F , the Galois groups G p m F : = Gal ( F p ( m ) / F ) ≃ G / G ( m ) of the finite stages F p ( m ) 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 G p m F of F with a fixed integer m ≥ 0 , namely the bound n ≤ m .

Theorem 3.2 (Successive Approximation Theorem.)

Let F be a number field, p a prime, and m,n integers. The abelian invariant collection τ ( ∞ ) F of F is approximated successively by the iterated AQI of sufficiently high p-class groups of F:

( ∀ n ≥ 1 ) ( ∀ m ≥ n ) τ ( n ) ( G p m F ) = τ ( n ) F . (3.11)

However, the transfer kernel type is a phenomenon of second order:

( ∀ m ≥ 2 ) ϰ ( G p m F ) ~ ϰ ( F ) , (3.12)

in particular, the metabelian second p-class group M : = G p 2 F ≃ G / G ″ of F is sufficient for determining the transfer kernel type of F.

Proof. This is one of the main results in ( [

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 m = n − 1 .

Conjecture 3.1 (Stage Separation Criterion.)

Denote by l p F the length of the p-class tower of F, that is the derived length dl ( G p ∞ F ) 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:

( ∀ n ≥ 1 ) l p F ≥ n ⇔ τ ( n ) ( G p n − 1 F ) < τ ( n ) F . (3.13)

The sufficiency of the condition in Conjecture 3.1 is a proven theorem ( [

Our first attempt to find sound asymptotic tendencies in the distribution of higher non-abelian p-class groups G p n F = Gal ( F p ( n ) / F ) , with n ≥ 2 , among the finite p-groups was planned in 1991 already ( [

Throughout this paper, isomorphism classes of finite groups G are characterized uniquely by their identifier in the SmallGroups Database [

For the decision if the p-class tower of a number field F is trivial with length l p F = 0 it suffices to compute the class number h ( F ) of the field.

Theorem 4.1 (Trivial p-class tower.)

The p-class tower of a number field F is trivial, F p ( ∞ ) = F , with length l p F = 0 , if and only if the class number h ( F ) = # Cl ( F ) is not divisible by p, i. e., the p-class number is h p F = 1 .

Proof. The proof consists of a sequence of equivalent statements: The class number satisfies p ∤ h ( F ) . Û The p-valuation of h ( F ) is v p ( h ( F ) ) = 0 . Û The p-class number is # Cl p F = h p F = p v p ( h ( F ) ) = 1 . Û The p-class group

Cl p F = 1 is trivial. Û The p-class rank ρ p = d i m F p ( Cl ( F ) / Cl ( F ) p ) is equal

to zero. Û The number of unramified cyclic extensions E / F of degree p is

p ρ p − 1 p − 1 = p 0 − 1 p − 1 = 1 − 1 p − 1 = 0 . Û The maximal unramified p-extension F p ( ∞ ) of F

coincides with F. Û The Galois group G p ∞ F = Gal ( F p ( ∞ ) / F ) = Gal ( F / F ) = 1 is trivial. Û The length of the p-class tower is l p F = dl ( G p ∞ F ) = dl ( 1 ) = 0 .

Already C. F. Gauss was able to compute class numbers h ( F ) of quadratic fields F = ℚ ( d ) , 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 Cl ( F ) . In 1998, Jacobson [

Corollary 4.1 (Statistics for p = 3 .) The asymptotic proportion of imaginary quadratic fields F = ℚ ( d ) , with negative discriminants d < 0 , whose class number h ( F ) is, respectively is not, divisible by p = 3 is given as 43.99%, respectively 56.01%, by the heuristics of Cohen, Lenstra and Martinet. In

Proof. The heuristic asymptotic limits are given in ( [

The first stage F p ( 1 ) of the p-class tower of a number field F is determined by the structure of the p-class group Cl p F of F as a finite abelian p-group. This is exactly the first order Artin pattern

AP ( 1 ) F = ( τ ( 1 ) F , ϰ ( 1 ) F ) = ( ATI ( Cl p F ) , k e r ( T F , F ) ) , (4.1)

since the trivial k e r ( T F , F ) = 1 does not contain information. However, only in the case of p-class rank one, ρ p = d i m F p ( Cl ( F ) / Cl ( F ) p ) = 1 , it is warranted that the exact length of the tower is l p F = 1 . A statistical example ( [

Theorem 4.2 A number field F with non-trivial cyclic p-class group Cl p F has an abelian p-class tower of exact length l p F = 1 , in fact, the Galois group G p ∞ F ≃ G p 1 F ≃ Cl p F is cyclic.

Proof. Suppose that Cl p F > 1 is non-trivial and cyclic. If the p-class tower had a length l p F ≥ 2 , the second p-class group M = G p 2 F would be a

L | # ( 3 | h ( F ) ) | rel. fr. | # ( 3 ∤ h ( F ) ) | rel. fr. | w. r. t. #total |
---|---|---|---|---|---|

−10^{6} | 121645 | 40.02% | 182323 | 59.98% | 303968 |

−10^{11} | 13206088529 | 43.45% | 17190266523 | 56.55% | 30396355052 |

−10^{12} | 132584350621 | 43.62% | 171379200091 | 56.38% | 303963550712 |

Cl 3 F ≃ | abs. fr. | rel. fr. | w. r. t. # ( ρ 3 = 1 ) |
---|---|---|---|

C 3 | 80115 | 67.63% | 118455 |

C 9 | 26458 | 22.34% | 118455 |

C 27 | 8974 | 7.58% | 118455 |

C 81 | 2472 | 2.09% | 118455 |

C 243 | 393 | 0.33% | 118455 |

C 729 | 43 | 0.04% | 118455 |

non-abelian finite p-group with cyclic abelianization M / M ′ ≃ Cl p F . However, it is well known that a nilpotent group with cyclic abelianization is abelian, which contradicts the assumption of a length l p F ≥ 2 .

Remark 4.1 We interpret the computation of abelian type invariants τ ( 1 ) F of the Sylow 3-subgroup Cl 3 F of the ideal class group Cl ( F ) of a quadratic field F = ℚ ( d ) as the determination of the single-stage approximation G / G ′ ≃ G 3 1 F ≃ Cl 3 F of the 3-class tower group G = G 3 ∞ F of F. This step yields complete information about the lattice of all unramified abelian 3-extensions E / F within the Hilbert 3-class field F 3 ( 1 ) of F.

According to the Successive Approximation Theorem 3.2, the second stage F p ( 2 ) of the p-class tower of a number field F is determined by the second order Artin pattern

AP ( 2 ) F = ( τ ( 2 ) F , ϰ ( 2 ) F ) = ( [ ATI ( Cl p F ) ; ( ATI ( Cl p E ) ) E ∈ Lyr 1 F ] , [ k e r ( T F , F ) ; ( k e r ( T F , E ) ) E ∈ Lyr 1 F ] ) . (4.2)

The determination of AP ( 2 ) F for a quadratic field F with 3-class rank ρ 3 = 2 requires the computation of four 3-class groups Cl 3 E i of unramified cyclic cubic extensions E 1 , ⋯ , E 4 and of four transfer kernels k e r ( T F , E i ) .

Whereas Mosunov and Jacobson [

Therefore, it must not be underestimated that Boston, Bush and Hajir [

Imaginary quadratic fields F = ℚ ( d ) with negative discriminants d < 0 are the simplest number fields with respect to their unit group U F , 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 [

Theorem 4.3 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist only two metabelian groups with GI-action (generator inverting action). and relation rank d 2 G = 2 (so-called Schur s-groups [

1) These are the groups of smallest order which are admissible as 3-class tower groups G ≃ G 3 ∞ F of imaginary quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 21,21,1 3 ,21 ) ] , [ 1 ; ( 2241 ) ] ) then G 3 ∞ F ≃ 〈 243,5 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 1 3 ,21,1 3 ,21 ) ] , [ 1 ; ( 4224 ) ] ) then G 3 ∞ F ≃ 〈 243,7 〉 .

3) The actual distribution of these 3-class tower groups G among the 276375 imaginary quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants − 10 8 < d < 0 is presented in

Proof. All finite 3-groups G with abelianization G / G ′ ≃ C 3 × C 3 are vertices of the descendant tree T ( R ) with abelian root R = 〈 9,2 〉 ≃ C 3 × C 3 . A search for metabelian vertices with relation rank d 2 G = 2 in this tree yields three hits 〈 27,4 〉 , 〈 243,5 〉 , and 〈 243,7 〉 , but only the latter two of them possess a GI-action.

The abelianization G / G ′ of a finite 3-group G which is realized as the 3-class tower group G p ∞ F of an algebraic number field F is isomorphic to the 3-class group Cl 3 F of F. When F is imaginary quadratic, it possesses signature ( r 1 , r 2 ) = ( 0,1 ) and torsionfree Dirichlet unit rank r = r 1 + r 2 − 1 = 0 . If G / G ′ ≃ Cl 3 F ≃ C 3 × C 3 , then the generator rank of G is d 1 G = 2 and the Shafarevich theorem implies bounds for the relation rank 2 = d 1 G ≤ d 2 G ≤ d 1 G + r = 2 .

The entries of

More recently, Boston, Bush and Hajir [

Real quadratic fields F = ℚ ( d ) with positive discriminants d > 0 are the second simplest number fields with respect to their unit group U F , 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 ( [

Theorem 4.4 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many

G ≃ | abs. fr. | rel. fr. | w. r. t. | rel. fr. | w. r. t. | measure [ | | d | min |
---|---|---|---|---|---|---|---|

〈 243,5 〉 | 83353 | 30.16% | 276375 | 18.04% | 461925 | 128 / 729 ≈ 17.56 % | 4027 |

〈 243,7 〉 | 41398 | 14.98% | 276375 | 8.96% | 461925 | 64 / 729 ≈ 8.78 % | 12131 |

metabelian groups with RI-action and relation rank d 2 G = 3 (so-called Schur + 1 s-groups [^{4}, namely 〈 81,7 〉 , 〈 81,8 〉 and 〈 81,10 〉 .

1) These are the groups of smallest order which are admissible as 3-class tower groups G ≃ G 3 ∞ F of real quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 1 3 ,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 81,7 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 21,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 81,8 〉 .

(c) If AP ( 2 ) F = ( [ 1 2 ; ( 21,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 1000 ) ] ) then G 3 ∞ F ≃ 〈 81,10 〉 .

3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 0 < d < 10 9 is presented in

Proof. A search for metabelian vertices G of minimal order with relation rank d 2 G = 3 in the descendant tree T ( R ) with abelian root R = 〈 9,2 〉 ≃ C 3 × C 3 yields three hits 〈 81,7 〉 , 〈 81,8 〉 , and 〈 81,10 〉 . All of them possess a RI-action.

The abelianization G / G ′ of a finite 3-group G which is realized as the 3-class tower group G p ∞ F of an algebraic number field F is isomorphic to the 3-class group Cl 3 F of F. When F is real quadratic, it possesses signature ( r 1 , r 2 ) = ( 2,0 ) and torsionfree Dirichlet unit rank r = r 1 + r 2 − 1 = 1 . If G / G ′ ≃ Cl 3 F ≃ C 3 × C 3 , then the generator rank of G is d 1 G = 2 and the Shafarevich theorem implies bounds for the relation rank 2 = d 1 G ≤ d 2 G ≤ d 1 G + r = 3 .

The entries of

In [

G ≃ | abs. fr. | rel. fr. | w. r. t. | rel. fr. | w. r. t. | measure [ | d min |
---|---|---|---|---|---|---|---|

〈 81,7 〉 | 122955 | 29.58% | 415698 | 25.52% | 481756 | 1664 / 6561 ≈ 25.36 % | 142097 |

〈 81,8 〉 or | 208236 | 50.09% | 415698 | 43.22% | 481756 | 8320 / 19683 ≈ 42.27 % | 32009 |

〈 81,10 〉 | |||||||

〈 243,5 〉 | 13712 | 3.30% | 415698 | 2.85% | 481756 | 1664 / 59049 ≈ 2.82 % | 422573 |

〈 243,7 〉 | 6691 | 1.61% | 415698 | 1.39% | 481756 | 832 / 59049 ≈ 1.41 % | 631769 |

patterns, for 0 < d < 10 7 in [

Corollary 4.2

1) If AP ( 2 ) F = ( [ 1 2 ; ( 32,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 1000 ) ] ) then G 3 ∞ F ≃ 〈 729,96 〉 .

2) If AP ( 2 ) F = ( [ 1 2 ; ( 32,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 729, i 〉 with i ∈ { 97,98 } .

3) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 0000 ) ] ) then G 3 ∞ F ≃ 〈 729, i 〉 with i ∈ { 99,100,101 } .

The actual distribution of these 3-class tower groups G among the 34631, respectively 2576, real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 0 < d < 10 8 , respectively 0 < d < 10 7 , is presented in

According to the Successive Approximation Theorem 3.2, the third stage F p ( 3 ) of the p-class tower of a number field F is usually determined by the third order Artin pattern

AP ( 3 ) F = ( τ ( 3 ) F , ϰ ( 3 ) F ) = ( [ τ ( 1 ) F ; ( τ ( 2 ) E ) E ∈ Lyr 1 F ] , [ ϰ ( 1 ) F ; ( ϰ ( 2 ) E ) E ∈ Lyr 1 F ] ) . (4.3)

It is interesting, however, that there are extensive collections of quadratic fields F with 3-class towers of exact length l 3 F = 3 , which can be characterized by the second order Artin pattern already. We begin with imaginary quadratic fields F = ℚ ( d ) with discriminants d < 0 .

Theorem 4.5 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many non-

G ≃ | abs. fr. | rel. fr. | w. r. t. | d min |
---|---|---|---|---|

〈 81 , 7 〉 | 10244 | 29.58% | 34631 | 142097 |

〈 81 , 8 〉 | 10514 | 30.36% | 34631 | 32009 |

〈 81 , 10 〉 | 7104 | 20.51% | 34631 | 72329 |

〈 729,96 〉 | 242 | 0.70% | 34631 | 790085 |

〈 729,97 〉 or | 713 | 2.06% | 34631 | 494236 |

〈 729,98 〉 | ||||

〈 729,99 〉 | 66 | 2.56% | 2576 | 62501 |

〈 729,100 〉 | 42 | 1.63% | 2576 | 152949 |

〈 729,101 〉 | 42 | 1.63% | 2576 | 252977 |

metabelian groups with GI-action and relation rank d 2 G = 2 (so-called Schur s-groups [^{8}, namely 〈 6561, i 〉 with i ∈ { 606,616,617,618,620,622,624 } .

1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups G ≃ G 3 ∞ F of imaginary quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

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 AP ( 2 ) F = ( [ 1 2 ; ( 32,21,1 3 ,21 ) ] , [ 1 ; ( 1313 ) ] ) then G 3 ∞ F ≃ 〈 6561,616 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,1 3 ,21 ) ] , [ 1 ; ( 2313 ) ] ) then G 3 ∞ F ≃ 〈 6561, i 〉 with i ∈ { 617,618 } .

(c) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,21,21 ) ] , [ 1 ; ( 1231 ) ] ) then G 3 ∞ F ≃ 〈 6561,622 〉 .

(d) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,21,21 ) ] , [ 1 ; ( 2231 ) ] ) then G 3 ∞ F ≃ 〈 6561, i 〉 with i ∈ { 620,624 } .

3) The actual distribution of these 3-class tower groups G among the 24476 imaginary quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants − 10 7 < d < 0 is presented in

Proof. By a similar but more extensive search than in the proof of Theorem 4.3. Data for

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 ( [

The group 〈 6561,606 〉 belongs to the infinite Shafarevich cover of the metabelian group 〈 729,45 〉 with respect to imaginary quadratic fields ( [

Now we turn to real quadratic fields F = ℚ ( d ) with discriminants d > 0 .

Theorem 4.6 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many non-

G ≃ | abs. fr. | rel. fr. | w. r. t. | type | ϰ | | d | min |
---|---|---|---|---|---|---|

〈 6561,616 〉 | 760 | 3.11% | 24476 | E.6 | (1313) | 15544 |

〈 6561,617 〉 or | 1572 | 6.42% | 24476 | E.14 | (2313) | 16627 |

〈 6561,618 〉 | ||||||

〈 6561,622 〉 | 798 | 3.26% | 24476 | E.8 | (1231) | 34867 |

〈 6561,620 〉 or | 1583 | 6.47% | 24476 | E.9 | (2231) | 9748 |

〈 6561,624 〉 |

metabelian groups with RI-action and relation rank d 2 G = 3 (so-called Schur + 1 s-groups [^{7}, namely 〈 2187, i 〉 with i ∈ { 270,271,272,273,284,291,307,308,311 } .

1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups G ≃ G 3 ∞ F of real quadratic fields F with 3- class group Cl 3 F ≃ C 3 × C 3 .

2) Exceptionally, for a real quadratic field F, four of these groups are determined by the second order Artin pattern already.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,21,1 3 ,21 ) ] , [ 1 ; ( 0313 ) ] ) then G 3 ∞ F ≃ 〈 2187, i 〉 with i ∈ { 284,291 } .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,21,21,21 ) ] , [ 1 ; ( 0231 ) ] ) then G 3 ∞ F ≃ 〈 2187, i 〉 with i ∈ { 307,308 } .

3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 1 < d < 10 9 is presented in

Proof. The claims for transfer kernel type c.18, ϰ ( F ) ~ ( 0313 ) , are a consequence of ( [

Remark 4.3 The groups 〈 2187, i 〉 with i ∈ { 270,271,272,273 } are elements of the infinite Shafarevich cover of the metabelian group 〈 729,45 〉 with respect to real quadratic fields.

The group 〈 2187,311 〉 belongs to the infinite Shafarevich cover of the metabelian group 〈 729,57 〉 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 ( [

Let ( γ i ( G ) ) i ≥ 1 be the descending lower central series of the group G, defined recursively by γ 1 ( G ) : = G and γ i ( G ) : = [ γ i − 1 ( G ) , G ] for i ≥ 2 , in particular, γ 2 ( G ) = G ′ is the commutator subgroup of G. A finite p-group G is nilpotent with γ 1 ( G ) > γ 2 ( G ) > ⋯ > γ c ( G ) > γ c + 1 ( G ) = 1 for some integer c ≥ 1 , which is

G ≃ | abs. fr. | rel. fr. | w. r. t. | type | ϰ | d min |
---|---|---|---|---|---|---|

〈 2187,284 〉 or | 4318 | 1.04% | 415698 | c.18 | (0313) | 534824 |

〈 2187,291 〉 | ||||||

〈 2187,307 〉 or | 4377 | 1.05% | 415698 | c.21 | (0231) | 540365 |

〈 2187,308 〉 |

called the nilpotency class cl ( G ) = c of G. When G is of order p n , for some integer n ≥ 1 , the coclass of G is defined by cc ( G ) : = n − c and lo ( G ) : = n is called the logarithmic order of G.

Finite 3-groups G with coclass cc ( G ) = 1 were investigated by N. Blackburn [

For the statement of Theorem 5.1, we need a precise ordering of the four maximal subgroups H 1 , ⋯ , H 4 of the group G = 〈 x , y 〉 , which can be generated by two elements x , y , according to the Burnside basis theorem. For this purpose, we select the generators x , y such that

H 1 = 〈 y , G ′ 〉 , H 2 = 〈 x , G ′ 〉 , H 3 = 〈 x y , G ′ 〉 , H 4 = 〈 x y 2 , G ′ 〉 , (5.1)

and H 1 = χ 2 ( G ) , provided that G is of nilpotency class cl ( G ) ≥ 3 . Here we denote by

χ 2 ( G ) : = { g ∈ G | ( ∀ h ∈ γ 2 ( G ) ) [ g , h ] ∈ γ 4 ( G ) } (5.2)

the two-step centralizer of G ′ in G.

Parametrized Presentations of Metabelian 3-GroupsThe identification of the groups will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn [

G a n ( z , w ) : = 〈 x , y , s 2 , ⋯ , s n − 1 | s 2 = [ y , x ] , ( ∀ i = 3 n ) s i = [ s i − 1 , x ] , s n = 1 , [ y , s 2 ] = s n − 1 a , ( ∀ i = 3 n − 1 ) [ y , s i ] = 1 , x 3 = s n − 1 w , y 3 s 2 3 s 3 = s n − 1 z , ( ∀ i = 2 n − 3 ) s i 3 s i + 1 3 s i + 2 = 1 , s n − 2 3 = s n − 1 3 = 1 〉 , (5.3)

where a ∈ { 0,1 } and w , z ∈ { − 1,0,1 } are bounded parameters, and the index of nilpotency n = cl ( G ) + 1 = cl ( G ) + cc ( G ) = l o g 3 ( ord ( G ) ) = : lo ( G ) is an unbounded parameter.

The following lemma generalizes relations for second and third powers of generators in ( [

Lemma 5.1 Let G = 〈 x , y 〉 be a finite 3-group with two generators x , y ∈ G . Denote by s 2 : = [ y , x ] the main commutator, and by s 3 : = [ s 2 , x ] and t 3 : = [ s 2 , y ] the two iterated commutators. Then the second and third power of the element x y , respectively x y 2 , are given by

( x y ) 2 = x 2 y 2 s 2 t 3 and ( x y ) 3 = x 3 y 3 s 2 3 s 3 t 3 2 , respectively ( x y 2 ) 2 = x 2 y 4 s 2 2 t 3 2 and ( x y 2 ) 3 = x 3 y 6 s 2 6 s 3 2 t 3 2 , (5.4)

provided that t 3 ∈ ζ ( G ) is central, t 3 3 = 1 , and [ s 3 , y ] = 1 .

Proof. We begin by preparing three commutator relations:

y x = x y [ y , x ] = x y s 2 , s 2 x = x s 2 [ s 2 , x ] = x s 2 s 3 and s 2 y = y s 2 [ s 2 , y ] = y s 2 t 3 . (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 t 3 belongs to the centre, t 3 3 = 1 , and s 3 y = y s 3 commute:

( x y ) 2 = x y x y = x x y s 2 y = x 2 y y s 2 t 3 = x 2 y 2 s 2 t 3 , and thus

( x y ) 3 = ( x y ) 2 x y = x 2 y 2 s 2 t 3 x y = x 2 y 2 s 2 x y t 3 = x 2 y y x s 2 s 3 y t 3 = x 2 y x y s 2 s 2 y s 3 t 3 = x 2 x y s 2 y s 2 y s 2 t 3 s 3 t 3 = x 3 y y s 2 t 3 y s 2 t 3 s 2 s 3 t 3 2 = x 3 y 2 s 2 y s 2 s 2 s 3 t 3 4 = x 3 y 2 y s 2 t 3 s 2 2 s 3 t 3 = x 3 y 3 s 2 3 s 3 t 3 2 , respectively

( x y 2 ) 2 = x y y x y y = x y x y s 2 y y = x x y s 2 y y s 2 t 3 y = x 2 y y s 2 t 3 y s 2 y t 3 = x 2 y 2 s 2 y y s 2 t 3 t 3 2 = x 2 y 2 y s 2 t 3 y s 2 t 3 3 = x 2 y 3 s 2 y s 2 t 3 = x 2 y 3 y s 2 t 3 s 2 t 3 = x 2 y 4 s 2 2 t 3 2 , and thus

( x y 2 ) 3 = ( x y 2 ) 2 x y 2 = x 2 y 4 s 2 2 t 3 2 x y 2 = x 2 y 4 s 2 s 2 x y y t 3 2 = x 2 y 4 s 2 x s 2 s 3 y y t 3 2 = x 2 y y y y x s 2 s 3 s 2 y y s 3 t 3 2 = x 2 y y y x y s 2 s 2 y s 2 t 3 y s 3 2 t 3 2 = x 2 y y x y s 2 y s 2 y s 2 t 3 s 2 y s 3 2 t 3 3 = x 2 y x y s 2 y y s 2 t 3 y s 2 t 3 s 2 y s 2 t 3 s 3 2 t 3 4 = x 2 x y s 2 y y s 2 t 3 y s 2 y s 2 t 3 2 y s 2 t 3 s 2 s 3 2 t 3 2 = x 3 y y s 2 t 3 y s 2 y y s 2 t 3 s 2 t 3 3 y s 2 2 s 3 2 t 3 3 = x 3 y 2 s 2 y s 2 y y s 2 t 3 2 s 2 y s 2 2 s 3 2

= x 3 y 2 y s 2 t 3 y s 2 t 3 y s 2 t 3 2 y s 2 t 3 s 2 2 s 3 2 = x 3 y 3 s 2 y s 2 t 3 2 y s 2 y t 3 3 s 2 3 s 3 2 = x 3 y 3 y s 2 t 3 s 2 t 3 2 y y s 2 t 3 s 2 3 s 3 2 = x 3 y 4 s 2 s 2 y y s 2 t 3 4 s 2 3 s 3 2 = x 3 y 4 s 2 s 2 y y s 2 4 s 3 2 t 3 = x 3 y 4 s 2 y s 2 t 3 y s 2 4 s 3 2 t 3 = x 3 y 4 s 2 y s 2 y s 2 4 s 3 2 t 3 2 = x 3 y 4 y s 2 t 3 y s 2 t 3 s 2 4 s 3 2 t 3 2 = x 3 y 5 s 2 y s 2 t 3 2 s 2 4 s 3 2 t 3 2 = x 3 y 5 y s 2 t 3 s 2 5 s 3 2 t 3 4 = x 3 y 6 s 2 6 s 3 2 t 3 2 .

Theorem 5.1 Let G = 〈 x , y 〉 ≃ G a n ( z , w ) be a finite 3-group of coclass cc ( G ) = 1 and order | G | = 3 n with generators x , y such that y ∈ χ 2 ( G ) is contained in the two-step centralizer of G, whereas x ∈ G \ χ 2 ( G ) , given by a polycyclic power commutator presentation with parameters a ∈ { 0,1 } , w , z ∈ { − 1,0,1 } , and index of nilpotency n ≥ 4 .

Then three of the four maximal subgroups, H i = 〈 x y i − 2 , G ′ 〉 < G , 2 ≤ i ≤ 4 , are non-abelian 3-groups of coclass cc ( H i ) = 1 , as listed in

The supplementary

Proof. For an index of nilpotency n ≥ 4 , the first maximal subgroup

G ≃ | n | a | z | w | H 2 = 〈 x , G ′ 〉 | H 3 = 〈 x y , G ′ 〉 | H 4 = 〈 x y 2 , G ′ 〉 |
---|---|---|---|---|---|---|---|

G 0 n ( 0,0 ) | ³4 | 0 | 0 | 0 | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,0 ) |

G 0 n ( 0,1 ) | ³4 | 0 | 0 | 1 | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) |

G 0 n ( 1,0 ) | ³4 | 0 | 1 | 0 | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) |

G 0 n ( − 1,0 ) | ³4 | 0 | −1 | 0 | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) |

G 1 n ( 0, − 1 ) | ³5 | 1 | 0 | −1 | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,0 ) |

G 1 n ( 0,0 ) | ³5 | 1 | 0 | 0 | ≃ G 0 n − 1 ( 0,0 ) | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) |

G 1 n ( 0,1 ) | ³5 | 1 | 0 | 1 | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) | ≃ G 0 n − 1 ( 0,1 ) |

G ≃ | n | a | z | w | H 1 = 〈 y , G ′ 〉 | H 2 = 〈 x , G ′ 〉 | H 3 = 〈 x y , G ′ 〉 | H 4 = 〈 x y 2 , G ′ 〉 |
---|---|---|---|---|---|---|---|---|

G 0 3 ( 0,0 ) | 3 | 0 | 0 | 0 | ≃ C 3 × C 3 | ≃ C 3 × C 3 | ≃ C 3 × C 3 | ≃ C 3 × C 3 |

G 0 3 ( 0,1 ) | 3 | 0 | 0 | 1 | ≃ C 3 × C 3 | ≃ C 9 | ≃ C 9 | ≃ C 9 |

H 1 = 〈 y , G ′ 〉 of G coincides with the two-step centralizer χ 2 ( G ) of G, which is a nearly homocyclic abelian 3-group A ( 3, n − 1 ) of order 3 n − 1 , when a = 0 . For a = 1 , we have H 1 / H ′ 1 ≃ A ( 3, n − 1 ) .

We transform all relations of the group G ≃ G a n ( z , w ) into relations of the remaining three maximal subgroups H ≃ G α n − 1 ( ζ , ω ) of G.

The polycyclic commutator relations s 2 = [ y , x ] , s i = [ s i − 1 , x ] for 3 ≤ i ≤ n , and the nilpotency relation s n = 1 for the group G = 〈 x , y 〉 , with lower central series γ i G = 〈 s i , γ i + 1 G 〉 for i ≥ 2 , can be used immediately for the subgroup H 2 = 〈 x , G ′ 〉 = 〈 x , s 2 〉 with lower central series γ i H 2 = 〈 t i , γ i + 1 H 2 〉 , where t i : = s i + 1 for i ≥ 2 , and t n − 1 = 1 .

For the lower central series of H 3 = 〈 x y , G ′ 〉 and H 4 = 〈 x y 2 , G ′ 〉 , we must employ the main commutator relation [ y , s 2 ] = s n − 1 a , and [ y , s i ] = 1 for i ≥ 3 . According to the right product rule for commutators, we have [ s i − 1 , x y ] = [ s i − 1 , y ] ⋅ [ s i − 1 , x ] y = 1 ⋅ s i y = s i [ s i , y ] = s i ⋅ 1 = s i , for i ≥ 4 , but [ s 2 , x y ] = [ s 2 , y ] ⋅ [ s 2 , x ] y = s n − 1 − a s 3 y = s n − 1 − a s 3 [ s 3 , y ] = s n − 1 − a s 3 , and in a similar fashion [ s i − 1 , x y 2 ] = [ s i − 1 , y ] ⋅ [ s i − 1 , x y ] y = 1 ⋅ s i y = s i [ s i , y ] = s i ⋅ 1 = s i , for i ≥ 4 , but again exceptionally [ s 2 , x y 2 ] = [ s 2 , y ] ⋅ [ s 2 , x y ] y = s n − 1 − a y − 1 s n − 1 − a s 3 y = s n − 1 − 2 a s 3 = s n − 1 a s 3 . For a = 1 , the left product rule for commutators shows [ s n − 1 ∓ 1 s 3 , x y ± 1 ] = [ s n − 1 ∓ 1 , x y ± 1 ] s 3 ⋅ [ s 3 , x y ± 1 ] = s 4 , that is, the slight anomaly for the main commutator disappears in the next step. Thus, the lower central series is γ i H j = 〈 t i , γ i + 1 H j 〉 for i ≥ 2 , 3 ≤ j ≤ 4 , where generally t i : = s i + 1 for i ≥ 3 , and t 2 : = s 3 for a = 0 , t 2 : = s n − 1 2 − j s 3 for a = 1 . In particular, H 3 = 〈 x y , s 2 〉 and H 4 = 〈 x y 2 , s 2 〉 .

The main commutator relation for all three subgroups H 2 , H 3 , H 4 of any group G ≃ G a n ( z , w ) with n ≥ 4 is [ s 2 , t 2 ] = 1 = t n − 2 α , that is α = 0 , generally, and it remains to determine ζ , ω .

For this purpose, we come to the power relations of G, x 3 = s n − 1 w , y 3 s 2 3 s 3 = s n − 1 z , and s i 3 s i + 1 3 s i + 2 = 1 for i ≥ 2 , supplemented by (5.4): ( x y ) 3 = x 3 y 3 s 2 3 s 3 s n − 1 − 2 a = s n − 1 w s n − 1 z s n − 1 − 2 a and ( x y 2 ) 3 = x 3 ( y 3 s 2 3 s 3 ) 2 s n − 1 − 2 a = s n − 1 w s n − 1 2 z s n − 1 − 2 a , and we use these relations to determine ζ , ω in dependence on w , z , a . Generally, we have s 2 3 t 2 3 t 3 = s 2 3 s 3 3 s 4 = 1 for a = 0 , s 2 3 t 2 3 t 3 = s 2 3 s n − 1 3 ( 2 − j ) s 3 3 s 4 = s 2 3 s 3 3 s 4 = 1 for a = 1 , and thus uniformly ζ = 0 .

For G 0 n ( 0,0 ) , we uniformly have x 3 = ( x y ) 3 = ( x y 2 ) 3 = 1 , and thus ω = 0 for all three subgroups. For G 0 n ( 0,1 ) , we uniformly have

x 3 = ( x y ) 3 = ( x y 2 ) 3 = s n − 1 , and thus ω = 1 for all three subgroups. For G 0 n ( ± 1,0 ) , we have x 3 = 1 , but ( x y ) 3 = s n − 1 ± 1 , ( x y 2 ) 3 = s n − 1 ± 2 = s n − 1 ∓ 1 , and thus ω = 0 for H 2 but ω = 1 for H 3 , H 4 , since G 0 n ( 0, − 1 ) ≃ G 0 n ( 0,1 ) .

For G 1 n ( 0, − 1 ) , we have x 3 = s n − 1 − 1 , but ( x y ) 3 = ( x y 2 ) 3 = s n − 1 − 3 = 1 , and thus ω = 1 for H 2 but ω = 0 for H 3 , H 4 . For G 1 n ( 0,0 ) , we have x 3 = 1 , but ( x y ) 3 = ( x y 2 ) 3 = s n − 1 − 2 = s n − 1 , and thus ω = 0 for H 2 but ω = 1 for H 3 , H 4 . For G 1 n ( 0,1 ) , we have x 3 = s n − 1 , ( x y ) 3 = ( x y 2 ) 3 = s n − 1 − 1 , and thus ω = 1 for all three subgroups, again observing that G 0 n ( 0, − 1 ) ≃ G 0 n ( 0,1 ) .

The only 3-groups G of coclass cc ( G ) = 1 and order | G | = 3 3 are the two extra special groups G 0 3 ( 0,0 ) and G 0 3 ( 0,1 ) . Since t 2 = s 3 = 1 , all their four maximal subgroups, H 1 = 〈 y , s 2 〉 , H 2 = 〈 x , s 2 〉 , H 3 = 〈 x y , s 2 〉 , H 4 = 〈 x y 2 , s 2 〉 , are abelian. For w = z = 0 , s 2 is independent of the other generator, and H i ≃ C 3 × C 3 for 1 ≤ i ≤ 4 . However, for w = 1 , z = 0 , we have

x 3 = ( x y ) 3 = ( x y 2 ) 3 = s 2 , s 2 3 = 1 , and thus H 2 ≃ H 3 ≃ H 4 ≃ C 9 , whereas H 1 ≃ C 3 × C 3 .

Suppose that p is a prime, F is an algebraic number field with non-trivial p-class group Cl p F > 1 , 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 F p ( ∞ ) = F p ( n ) of exact length l p F = n for some integer n ≥ 1 . Then the Galois group Gal ( E p ( ∞ ) / E ) of the p-class tower of E is a subgroup of index E : F of the p-class tower group Gal ( F p ( ∞ ) / F ) of F and the length of the p-class tower of E is bounded by l p E ≤ n .

Proof. According to the assumptions, there exists a tower of field extensions,

F < E ≤ F p ( 1 ) ≤ E p ( 1 ) ≤ F p ( 2 ) ≤ E p ( 2 ) ≤ ⋯ ≤ F p ( n ) ≤ E p ( n ) ≤ F p ( n + 1 ) ,

where l p F = n enforces the coincidence F p ( n ) = E p ( n ) = F p ( n + 1 ) of the trailing three fields. Since Gal ( F p ( n ) / F ) / Gal ( F p ( n ) / E ) ≃ Gal ( E / F ) , the group index of Gal ( E p ( n ) / E ) = Gal ( F p ( n ) / E ) in Gal ( F p ( n ) / F ) is equal to the field degree [ E : F ] and Gal ( E p ( ∞ ) / E ) = Gal ( E p ( n ) / E ) is a subgroup of index [ E : F ] of Gal ( F p ( n ) / F ) = Gal ( F p ( ∞ ) / F ) . The equality E p ( n ) = E p ( n + 1 ) implies the bound l p E ≤ n .

We shall apply Theorem 6.1 to the situation where p = 3 , n = 2 , and E is an

unramified cyclic cubic extension of F, whence Gal ( E 3 ( ∞ ) / E ) is a maximal subgroup of Gal ( F 3 ( ∞ ) / F ) .

Proposition 6.1 Let G be a finite 3-group with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of G is D.10, ϰ ( G ) ~ ( 2241 ) .

2) The abelian quotient invariants of the four maximal subgroups H 1 , ⋯ , H 4 of G are τ ( G ) ~ ( 21,21,1 3 ,21 ) .

3) The isomorphism types of the four maximal subgroups of G are H 1 ≃ H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 and H 3 ≃ 〈 3 4 ,13 〉 .

4) The group G is isomorphic to the Schur s-group 〈 3 5 ,5 〉 with relation rank d 2 = 2 .

Proof. We put G : = 〈 243 , 5 〉 and use the presentation [

G = 〈 x , y , s 2 , s 3 , t 3 | s 2 = [ y , x ] , s 3 = [ s 2 , x ] , t 3 = [ s 2 , y ] , x 3 = s 3 , y 3 = s 3 〉 .

Then we obtain the maximal subgroups

H 1 = 〈 y , G ′ 〉 = 〈 y , s 2 , s 3 〉 , since t 3 = [ s 2 , y ] ,

H 2 = 〈 x , G ′ 〉 = 〈 x , s 2 , t 3 〉 , since s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , G ′ 〉 = 〈 x y , s 2 , s 3 〉 , since [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , G ′ 〉 = 〈 x y 2 , s 2 , s 3 〉 , since [ s 2 , x y 2 ] = s 3 t 3 2 .

Using Lemma 5.1, and comparing to the abstract presentations [

〈 81 , 3 〉 = 〈 ξ , υ , σ 2 , τ | σ 2 = [ υ , ξ ] , τ = ξ 3 〉 and

〈 81 , 13 〉 = 〈 ξ , υ , ζ , σ 2 | σ 2 = [ υ , ξ ] , ξ 3 = σ 2 , υ 3 = ζ 3 = 1 〉 ,

we conclude

H 1 = 〈 y , s 2 , s 3 〉 = 〈 y , s 2 〉 ≃ 〈 81 , 3 〉 , since y 3 = s 3 ≠ [ s 2 , y ] = t 3 ,

H 2 = 〈 x , s 2 , t 3 〉 ≃ 〈 81 , 13 〉 , since x 3 = s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , s 2 , s 3 〉 = 〈 x y , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y ) 3 = t 3 2 ≠ [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , s 2 , s 3 〉 = 〈 x y 2 , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y 2 ) 3 = s 3 2 t 3 2 ≠ [ s 2 , x y 2 ] = s 3 t 3 2 .

Theorem 6.2 Let F = ℚ ( d ) be a quadratic field with elementary bicyclic 3-class group Cl 3 F ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of F is D.10, ϰ ( F ) ~ ( 2241 ) .

2) The abelian type invariants of the 3-class groups Cl 3 E i of the four unramified cyclic cubic extensions E i / F are τ ( F ) ~ ( 21,21,1 3 ,21 ) .

3) The second 3-class group G 3 2 F of F has the maximal subgroups H 1 ≃ H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 and H 3 ≃ 〈 3 4 ,13 〉 .

4) The 3-class tower group G 3 ∞ F of F is the Schur s-group 〈 3 5 ,5 〉 with relation rank d 2 = 2 .

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 l 3 F = 2 . The four unramified cyclic cubic extensions E i / F are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank r ≥ 2 , and possess 3-class towers of length l 3 E i = 2 . More precisely, Cl 3 E 3 ≃ C 3 × C 3 × C 3 and G 3 ∞ E 3 ≃ 〈 3 4 ,13 〉 with relation rank d 2 = 5 , but Cl 3 E i ≃ C 9 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,3 〉 with relation rank d 2 = 4 for i ∈ { 1,2,4 } .

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 G / G ′ ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of G is D.5, ϰ ( G ) ~ ( 4224 ) .

2) The abelian quotient invariants of the four maximal subgroups H 1 , ⋯ , H 4 of G are τ ( G ) ~ ( 1 3 ,21,1 3 ,21 ) .

3) The isomorphism types of the four maximal subgroups of G are H 1 ≃ H 3 ≃ 〈 3 4 ,13 〉 and H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 .

4) The group G is isomorphic to the Schur s-group 〈 3 5 ,7 〉 with relation rank d 2 = 2 .

Proof. We put G : = 〈 243 , 7 〉 and use the presentation [

G = 〈 x , y , s 2 , s 3 , t 3 | s 2 = [ y , x ] , s 3 = [ s 2 , x ] , t 3 = [ s 2 , y ] , x 3 = s 3 , y 3 = s 3 2 〉 .

Similarly as in Proposition 6.1, we obtain the maximal subgroups

H 1 = 〈 y , G ′ 〉 = 〈 y , s 2 , s 3 〉 , H 2 = 〈 x , G ′ 〉 = 〈 x , s 2 , t 3 〉 ,

H 3 = 〈 x y , G ′ 〉 = 〈 x y , s 2 , s 3 〉 , and H 4 = 〈 x y 2 , G ′ 〉 = 〈 x y 2 , s 2 , s 3 〉 .

Using Lemma 5.1, and comparing to the abstract presentations

〈 81 , 3 〉 = 〈 ξ , υ , σ 2 , τ | σ 2 = [ υ , ξ ] , τ = ξ 3 〉 and

〈 81 , 13 〉 = 〈 ξ , υ , ζ , σ 2 | σ 2 = [ υ , ξ ] , ξ 3 = σ 2 , υ 3 = ζ 3 = 1 〉 ,

we conclude

H 1 = 〈 y , s 2 , s 3 〉 = 〈 y , s 2 〉 ≃ 〈 81 , 3 〉 , since y 3 = s 3 2 ≠ [ s 2 , y ] = t 3 ,

H 2 = 〈 x , s 2 , t 3 〉 ≃ 〈 81 , 13 〉 , since x 3 = s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , s 2 , s 3 〉 = 〈 x y , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y ) 3 = s 3 t 3 2 ≠ [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , s 2 , s 3 〉 ≃ 〈 81 , 13 〉 , since ( x y 2 ) 3 = s 3 t 3 2 = [ s 2 , x y 2 ] .

Theorem 6.3 Let F = ℚ ( d ) be a quadratic field with elementary bicyclic 3-class group Cl 3 F ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of F is D.5, ϰ ( F ) ~ ( 4224 ) .

2) The abelian type invariants of the 3-class groups Cl 3 E i of the four unramified cyclic cubic extensions E i / F are τ ( F ) ~ ( 1 3 ,21,1 3 ,21 ) .

3) The second 3-class group G 3 2 F of F has the maximal subgroups H 1 ≃ H 3 ≃ 〈 3 4 ,13 〉 and H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 .

4) The 3-class tower group G 3 ∞ F of F is the Schur s-group 〈 3 5 ,7 〉 with relation rank d 2 = 2 .

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 l 3 F = 2 . The four unramified cyclic cubic extensions E i / F are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank r ≥ 2 , and possess 3-class towers of length l 3 E i = 2 . More precisely, Cl 3 E i ≃ C 3 × C 3 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,13 〉 with relation rank d 2 = 5 for i ∈ { 1,3 } , but Cl 3 E i ≃ C 9 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,3 〉 with relation rank d 2 = 4 for i ∈ { 2,4 } .

Proof. This is a consequence of Theorems 6.1 and 6.3, satisfying the Shafarevich theorem.

We recall that a dihedral field E of degree 6 is an absolute Galois extension E / ℚ with group Gal ( E / ℚ ) = 〈 σ , τ | σ 3 = τ 2 = 1 , σ τ = τ σ − 1 〉 . It is a cyclic cubic relative extension E / F of its unique quadratic subfield F = E σ , and it contains three isomorphic, conjugate non-Galois cubic subfields L = E τ , L σ , L σ 2 . The conductor c of E / F is a nearly squarefree positive integer with special prime factors, and the discriminants satisfy the relations d E = c 4 d F 3 and d L = c 2 d F . Here, we shall always be concerned with unramified extensions, characterized by the conductor c = 1 , and thus d E = d F 3 , a perfect cube, and equal d L = d F .

The computational information on 3-tower groups G : = G 3 ∞ F of imaginary quadratic fields F in

The computational information on 3-tower groups G : = G 3 ∞ F of real quadratic fields F in

G ≃ | τ ( 1 ) G | abs. fr. | S ≃ | τ ( 1 ) S | abs. fr. | | d E | min |
---|---|---|---|---|---|---|

〈 243,5 〉 | 1^{2} | 83353 | 〈 81,3 〉 | 21 | 250059 | 4027^{3} |

〈 243,5 〉 | 1^{2} | 83353 | 〈 81,13 〉 | 1^{3} | 83353 | 4027^{3} |

〈 243,7 〉 | 1^{2} | 41398 | 〈 81,3 〉 | 21 | 82796 | 12131^{3} |

〈 243,7 〉 | 1^{2} | 41398 | 〈 81,13 〉 | 1^{3} | 82796 | 12131^{3} |

G ≃ | τ ( 1 ) G | abs. fr. | S ≃ | τ ( 1 ) S | abs. fr. | ( d E ) min |
---|---|---|---|---|---|---|

〈 81,7 〉 | 1^{2} | 122955 | 〈 27,3 〉 | 1^{2} | 122955 | 142097^{3} |

〈 81,7 〉 | 1^{2} | 122955 | 〈 27,4 〉 | 1^{2} | 245910 | 142097^{3} |

〈 81,7 〉 | 1^{2} | 122955 | 〈 27,5 〉 | 1^{2} | 122955 | 142097^{3} |

The first row of

Theorem 6.4 (A new realization as 3-tower group.) The extraspecial 3-group S = 〈 27 , 3 〉 of coclass 1 and exponent 3 occurs as 3-class tower group G 3 ∞ E of totally real dihedral fields E of degree 6.

Proof. The group S = 〈 27 , 3 〉 possesses the relation rank d 2 S = 4 . According to the Shafarevich Theorem, it is therefore excluded as 3-tower group G 3 ∞ F 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 G 3 ∞ E of totally real dihedral fields E of degree 6, as visualized in

Theorem 6.5 (3-class tower groups of totally real dihedral fields.) Let F = ℚ ( d ) be a real quadratic field with 3-class group Cl 3 F ≃ C 3 × C 3 and fundamental discriminant d > 1 . Suppose the second order Artin pattern AP ( 2 ) F = ( τ ( 2 ) ( F ) , ϰ ( 2 ) ( F ) ) is given by the abelian type invariants τ ( 2 ) ( F ) = [ 1 2 ; ( 2 2 ,1 2 ,1 2 ,1 2 ) ] and the transfer kernel type ϰ ( 2 ) ( F ) = [ 1 ; ( 0000 ) ] . Let E 2 , E 3 , E 4 be the three unramified cyclic cubic relative extensions of F with 3-class group Cl 3 E i ≃ C 3 × C 3 .

Then E i / ℚ is a totally real dihedral extension of degree 6, for each 2 ≤ i ≤ 4 ,

and the connection between the component # ϰ ( 3 ) ( F ) i = # k e r ( T E i , F 3 ( 1 ) ) of the third order transfer kernel type ϰ ( 3 ) ( F ) and the 3-class tower group S i = G 3 ∞ E i = Gal ( ( E i ) 3 ( ∞ ) / E i ) of E i is given in the following way:

# ϰ ( 3 ) ( F ) i = 3 ⇔ S i ≃ 〈 243,27 〉 with ϰ ( S i ) = ( 1000 ) , # ϰ ( 3 ) ( F ) i = 9 ⇔ S i ≃ 〈 243,26 〉 with ϰ ( S i ) = ( 0000 ) . (6.1)

Proof. This theorem was expressed as a conjecture in [

Remark 6.1 Recall that each unramified cyclic cubic relative extension E i / F , 1 ≤ i ≤ 4 , gives rise to a dihedral absolute extension E i / ℚ of degree 6, that is an S 3 -extension ( [

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 G 3 ∞ F of quadratic number fields F = ℚ ( d ) in Sections 4.2-4.5. In particular, the discovery of non-metabelian 3-class towers with exact length l 3 F = 3 , 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 [

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.

Mayer, D.C. (2017) Successive Approximation of p-Class Towers. Advances in Pure Mathematics, 7, 660-685. https://doi.org/10.4236/apm.2017.712041