Deep Transfers of p-Class Tower Groups

Daniel C. Mayer

Naglergasse 53, Graz, Austria.

DOI: 10.4236/jamp.2018.61005                

Abstract

Let p be a prime. For any finite p-group G, the deep transfers T _{H,G ^' : H / H ^' → G ^' / G ^" from the maximal subgroups H of index (G：H) = p in G to the derived subgroup G ^' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ^')) (G： H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 ^(∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 < d < 10⁷, and a few exceptional cases are pointed out for 1 < d < 10⁸.}

Keywords

Hilbert p-Class Field Towers, p-Class Groups, p-Principalization, Quadratic Fields, Dihedral Fields of Degree 2p; Finite p-Groups, Two-Step Centralizers, Polarization Principle, Descendant Trees, p-Group Generation Algorithm, p-Multiplicator Rank, Relation Rank, Generator Rank, Deep Transfers, Shallow Transfers, Partial Order and Monotony Principle of Artin Patterns, Parametrized Polycyclic pc-Presentations, Commutator Calculus

1. Introduction

The layout of this paper is the following. Deep transfers of finite p-groups G, with an assigned prime number p, are introduced as an innovative supplement to the (usual) shallow transfers [[1], p. 50], [[2], Equation (4), p. 470] in §2. The family ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{H\mathrm{,}{G}^{\prime }}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo>\; :\; </mo>}H\mathrm{<mo\; stretchy="false">\; )\; </mo>}=p}$ of the kernels of all deep transfers of G is called the deep transfer kernel type of G and will play a crucial role in this paper. For all finite 3-groups G of coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ , the deep transfer kernel type ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{H_i\mathrm{,}{G}^{\prime }}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{1\le i\le 4}$ is determined explicitly with the aid of commutator calculus in §3 using a parametrized polycyclic power-commutator presentation of G [3] [4] [5]. In the concluding §4, the orders of the deep transfer kernels are sufficient for identifying the Galois group $G_3^{\infty }F\mathrm{<mo>\; :\; </mo>}=\text{Gal}\mathrm{<mo\; stretchy="false">\; (\; </mo>}{F}_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\infty \mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of the maximal unramified pro-3 extension of real quadratic fields $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ , and total 3-principalization in each of their four unramified cyclic cubic extensions $E_1\mathrm{,}\dots \mathrm{,}{E}_4$ .

2. Shallow and Deep Transfer of p-Groups

With an assigned prime number $p\ge 2$ , let G be a finite p-group. Since our focus in this paper will be on the simplest possible non-trivial situation, we assume that the abelianization $G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }$ of G is of elementary type $\mathrm{<mo\; stretchy="false">\; (\; </mo>}p\mathrm{,}p\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with rank two. For applications in number theory, concerning p-class towers, the Artin pattern has proved to be a decisive collection of information on G.

Definition 2.1. The Artin pattern $\text{AP}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>}\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,}\varkappa \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}$ of G consists of two families

$\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>}{H}_i\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_{1\le i\le p+1}\text{and}\varkappa \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{G\mathrm{,}{H}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{1\le i\le p+1}$ (2.1)

containing the targets and kernels of the Artin transfer homomorphisms $T_{G,H_i}:G/G^{\prime }\to H_i/{H^{\prime }}_i$ [[5], Lem. 6.4, p. 198], [[2], Equation (4), p. 470] from G to its $p+1$ maximal subgroups $H_i$ with $i\in \mathrm{<mo>\; \{\; </mo>\; 1,}\dots \mathrm{,}p+\mathrm{1\; <mo>\; \}\; </mo>}$ . Since the maximal subgroups form the shallow layer ${\text{Lyr}}_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of subgroups of index $(G:H_i)=p$ of G, we shall call the $T_{G\mathrm{,}{H}_i}$ the shallow transfers of G, and ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}=\varkappa \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ the shallow transfer kernel type (sTKT) of G.

We recall [[2], §2.2, pp. 475-476] that the sTKT is usually simplified by a family of non-negative integers, in the following way. For $1\le i\le p+1$ ,

${\varkappa }_s{\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i\mathrm{<mo>\; :\; </mo>}=(\begin{array}{ll}j\hfill & \text{if}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{G\mathrm{,}{H}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>}=H_j\mathrm{<mo>\; /\; </mo>}{G}^{\prime }\text{for}\text{some}j\in \mathrm{<mo>\; \{\; </mo>\; 1,}\dots \mathrm{,}p+\mathrm{1\; <mo>\; \}\; </mo>\; ,}\hfill \\ 0\hfill & \text{if}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{G\mathrm{,}{H}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>}=G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }\mathrm{.}\hfill \end{array}$ (2.2)

The progressive innovation in this paper, however, is the introduction of the deep Artin transfer.

Definition 2.2. By the deep transfers we understand the Artin transfer homomorphisms $T_{H_i\mathrm{,}{G}^{\prime }}\mathrm{<mo>\; :\; </mo>}{H}_i\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_i\to G^{\prime }/G^{″}$ [[5], Lem. 6.1, p. 196], [[6], Dfn. 3.3, p. 69] from the maximal subgroups $H_1\mathrm{,}\dots \mathrm{,}{H}_{p+1}$ to the commutator subgroup $G^{\prime }$ of G, which forms the deep layer ${\text{Lyr}}_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of the (unique) subgroup of index $(G:G^{\prime })=p^2$ of G with abelian quotient $G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }$ . Accordingly, we call the family

${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>\; <mo>\; \#\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{H_i\mathrm{,}{G}^{\prime }}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{1\le i\le p+1}$ (2.3)

the deep transfer kernel type (dTKT) of G.

We point out that, as opposed to the sTKT, the members of the dTKT are only cardinalities, since this will suffice for reaching our intended goals in this paper. This preliminary coarse definition is open to further refinement in subsequent publications (See the proof of Theorem 3.1.).

3. Identification of 3-Groups by Deep Transfers

The drawback of the sTKT is the fact that occasionally several non-isomorphic p-groups G share a common Artin pattern $\text{AP}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>}\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,}{\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}$ [[7], Thm. 7.2, p. 158]. The benefit of the dTKT is its ability to distinguish the members of such batches of p-groups which have been inseparable up to now. After the general introduction of the dTKT for arbitrary p-groups in §2, we are now going to demonstrate its advantages in the particular situation of the prime $p=3$ and finite 3-groups G of coclass $cc(G)=1$ , which are necessarily metabelian with second derived subgroup $G^{″}=1$ and abelianization $G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }\simeq C_3\times C_3$ , according to Blackburn [8].

For the statement of our main theorem, we need a precise ordering of the four maximal subgroups $H_1\mathrm{,}\dots \mathrm{,}{H}_4$ of the group $G=\langle x,y\rangle$ , which can be generated by two elements $x\mathrm{,}y$ , according to the Burnside basis theorem. For this purpose, we select the generators $x\mathrm{,}y$ such that

$H_1=\langle y,G^{\prime }\rangle ,H_2=\langle x,G^{\prime }\rangle ,H_3=\langle xy,G^{\prime }\rangle ,H_4=\langle x{y}^2,G^{\prime }\rangle ,$ (3.1)

and $H_1={\chi }_2(G)$ , provided that G is of nilpotency class $\text{cl}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}\ge 3$ . Here we denote by

${\chi }_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; :\; </mo>}=\mathrm{<mo>\; \{\; </mo>}g\in G|\mathrm{<mo\; stretchy="false">\; (\; </mo>}\forall h\in G^{\prime }\mathrm{<mo\; stretchy="false">\; )\; </mo>}[g\mathrm{,}h]\in {\gamma }_4\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; \}\; </mo>}$ (3.2)

the two-step centralizer of $G^{\prime }$ in G, where we let ${\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\gamma }_i\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{i\ge 1}$ be the lower central series of $G=:{\gamma }_1(G)$ with ${\gamma }_i(G)=[{\gamma }_{i-1}(G),G]$ for $i\ge 2$ , in particular, ${\gamma }_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=G^{\prime }$ .

The identification of the groups will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn [3], Miech [4], and Nebelung [5]:

$\begin{array}{l}{G}_a^n(z,w):=\langle x,y,s_2,\dots ,s_{n-1}|s_2=y,x,({\forall }_{i=3}^n)s_i=[s_{i-1},x],s_n=1,[y,s_2]=s_{n-1}^a,\\ ({\forall }_{i=3}^{n-1})[y,s_i]=1,x^3=s_{n-1}^w,y^3{s}_2^3{s}_3=s_{n-1}^z,({\forall }_{i=2}^{n-3})s_i^3{s}_{i+1}^3{s}_{i+2}=1,s_{n-2}^3=s_{n-1}^3=1\rangle ,\end{array}$ (3.3)

where $a\in \mathrm{<mo>\; \{\; </mo>\; 0,1\; <mo>\; \}\; </mo>}$ and $w\mathrm{,}z\in \mathrm{<mo>\; \{\; </mo>}-\mathrm{1,0,1\; <mo>\; \}\; </mo>}$ are bounded parameters, and the index of nilpotency $n=\text{cl}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}+1=\text{cl}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}+\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=lo{g}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}\text{ord}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo>\; :\; </mo>}\text{lo}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ is an unbounded parameter.

Lemma 3.1. Let G be an arbitrary group with elements $x\mathrm{,}y\in G$ . Then the second and third power of the product $xy$ are given by

1) ${(xy)}^2=x^2{y}^2{s}_2{t}_3$ , where $s_2:=[y,x]$ , $t_3:=[s_2,y]$ ,

2) ${(xy)}^3=x^3{y}^3{(s_2{t}_3^2{t}_4)}^2{s}_3{u}_4^2{u}_5{s}_2{t}_3$ , where $s_3=[s_2,x]$ , $t_4=[t_3,y]$ , $u_4=[s_3,y]$ , $u_5=[u_4,y]$ .

If $G\simeq G_a^n\mathrm{<mo\; stretchy="false">\; (\; </mo>}z\mathrm{,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ , then ${(xy)}^2=x^2{y}^2{s}_2{s}_{n-1}^{-a}$ and ${(xy)}^3=x^3{y}^3{s}_2^3{s}_3{s}_{n-1}^{-2a}$ , and the second and third power of $x{y}^2$ are given by ${(x{y}^2)}^2=x^2{y}^4{s}_2^2{s}_{n-1}^{-2a}$ and ${(x{y}^2)}^3=x^3{y}^6{s}_2^6{s}_3^2{s}_{n-1}^{-2a}$ .

Proof. We prepare the calculation of the powers by proving a few preliminary identities:

$yx=1\cdot yx=xy{y}^{-1}{x}^{-1}\cdot yx=xy\cdot y^{-1}{x}^{-1}yx=xy\cdot [y,x]=xy{s}_2$ , and similarly $s_{2}y=y{s}_2\cdot [s_2,y]=y{s}_2{t}_3$ and $s_{2}x=x{s}_2\cdot [s_2,x]=x{s}_2{s}_3$ and $t_{3}y=y{t}_3\cdot [t_3,y]=y{t}_3{t}_4$ and $s_{3}y=y{s}_3\cdot [s_3,y]=y{s}_3{u}_4$ and $u_{4}y=y{u}_4\cdot [u_4,y]=y{u}_4{u}_5$ . Furthermore, $y{x}^2=yx\cdot x=xy{s}_2\cdot x=xy\cdot s_{2}x=xy\cdot x{s}_2{s}_3=x\cdot yx\cdot s_2{s}_3=x\cdot xy{s}_2\cdot s_2{s}_3=x^{2}y{s}_2^2{s}_3$ , $s_2{y}^2=s_{2}y\cdot y=y{s}_2{t}_3\cdot y=y{s}_2\cdot t_{3}y=y{s}_2\cdot y{t}_3{t}_4=y\cdot s_{2}y\cdot t_3{t}_4=y\cdot y{s}_2{t}_3\cdot t_3{t}_4=y^2{s}_2{t}_3^2{t}_4$ , $\begin{array}{c}{s}_3{y}^2=s_{3}y\cdot y=y{s}_3{u}_4\cdot y=y{s}_3\cdot u_{4}y=y{s}_3\cdot y{u}_4{u}_5\\ =y\cdot s_{3}y\cdot u_4{u}_5=y\cdot y{s}_3{u}_4\cdot u_4{u}_5=y^2{s}_3{u}_4^2{u}_5\end{array}$ .

Now the second power of $xy$ is

${(xy)}^2=xyxy=x\cdot yx\cdot y=x\cdot xy{s}_2\cdot y=x^{2}y\cdot s_{2}y=x^{2}y\cdot y{s}_2{t}_3=x^2{y}^2{s}_2{t}_3$

and the third power of $xy$ is

$\begin{array}{l}{(}^x=xy\cdot {(}^x=xy\cdot x^2{y}^2{s}_2{t}_3=x\cdot y{x}^2\cdot y^2{s}_2{t}_3=x\cdot x^{2}y{s}_2^2{s}_3\cdot y^2{s}_2{t}_3\\ =x^{3}y{s}_2^2\cdot s_3{y}^2\cdot s_2{t}_3=x^{3}y{s}_2^2\cdot y^2{s}_3{u}_4^2{u}_5\cdot s_2{t}_3=x^{3}y{s}_2\cdot s_2{y}^2\cdot s_3{u}_4^2{u}_5{s}_2{t}_3\\ =x^{3}y{s}_2\cdot y^2{s}_2{t}_3^2{t}_4\cdot s_3{u}_4^2{u}_5{s}_2{t}_3=x^{3}y\cdot s_2{y}^2\cdot s_2{t}_3^2{t}_4{s}_3{u}_4^2{u}_5{s}_2{t}_3\\ =x^{3}y\cdot y^2{s}_2{t}_3^2{t}_4\cdot s_2{t}_3^2{t}_4{s}_3{u}_4^2{u}_5{s}_2{t}_3=x^3{y}^3{(}^{s_2}{s}_3{u}_4^2{u}_5{s}_2{t}_3\end{array}$ .

If $G\simeq G_a^n\mathrm{<mo\; stretchy="false">\; (\; </mo>}z\mathrm{,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ , then $t_4=u_4=u_5=1$ , $t_3=s_{n-1}^{-a}$ , $t_3^3=s_{n-1}^{-3a}=1$ , and $G^{\prime }$ is abelian. □

Theorem 3.1. (3-groups G of coclass cc(G) = 1.) Let G be a finite 3-group of coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ and order $\text{ord}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=3^n$ with an integer exponent $n\ge 2$ . Then the shallow and deep transfer kernel type of G are given in dependence on the relational parameters $a\mathrm{,}n\mathrm{,}w\mathrm{,}z$ of $G\simeq G_a^n\mathrm{<mo\; stretchy="false">\; (\; </mo>}z\mathrm{,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ by Table 1.

Proof. The shallow TKT ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of all 3-groups G of coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ has been determined in [2], where the designations a.n of the types were introduced with $n\in \mathrm{<mo>\; \{\; </mo>\; 1,2,3\; <mo>\; \}\; </mo>}$ . Here, we indicate a capable mainline vertex of the tree ${\mathcal{T}}^1\mathrm{<mo\; stretchy="false">\; (\; </mo>}R\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with root $R=C_3\times C_3$ [7] by the type a.1* with a trailing asterisk. As usual, type a.3* indicates the unique 3-group $G\simeq {\text{Syl}}_3{A}_9$ with $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3,3\; <mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ . Now we want to determine the deep TKT ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ , using the presentation of $G\simeq G_a^n\mathrm{<mo\; stretchy="false">\; (\; </mo>}z\mathrm{,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ in Formula (3.3). For this purpose, we need expressions for the images of the deep Artin transfers $T_i\mathrm{<mo>\; :\; </mo>}=T_{H_i\mathrm{,}{G}^{\prime }}\mathrm{<mo>\; :\; </mo>}{H}_i\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_i\to G^{\prime }$ , for each $1\le i\le 4$ . (Observe that $p=3$ implies $G^{″}=1$ by [8].) Generally, we have to distinguish outer transfers, $T_i(g\cdot {H^{\prime }}_i)=g^3$ if $g\in H_i\backslash G^{\prime }$ [[2], Equation (4), p. 470], and inner transfers, $T_i(g\cdot {H^{\prime }}_i)=g^{1+h+h^2}=g^3\cdot [g,h^3]\cdot [[g,h],h]$ if $g\in G^{\prime }$ and h is selected in $H_i\backslash G^{\prime }$ [[2], Equation (6), p. 486].

First, we consider the distinguished two-step centralizer $H_1={\chi }_2(G)$ with $i=1$ . Then $H_1=\langle y,G^{\prime }\rangle$ and ${H^{\prime }}_1=1$ if $a=0$ ( $H_1$ abelian), but ${H^{\prime }}_1={\gamma }_{n-1}(G)=\langle s_{n-1}\rangle$ if $a=1$ ( $H_1$ non-abelian) [[2], Equation (3), p. 470]. The outer transfer is determined by $T_1(y\cdot {H^{\prime }}_1)=y^3=s_2^{-3}{s}_3^{-1}{s}_{n-1}^z$ . For the inner transfer, we have $T_1(s_j\cdot {H^{\prime }}_1)=s_j^{1+y+y^2}=s_j^3\cdot [s_j,y^3]\cdot [[s_j,y],y]=s_j^3\cdot 1^3\cdot [1,y]=s_j^3$ for all $j\ge 3$ , but $T_1(s_2\cdot {H^{\prime }}_1)=s_2^3\cdot s_{n-1}^{-3a}\cdot [s_{n-1}^{-a},y]=s_2^3$ for $j=2$ , since

$s_{n-1}^{-a}\in \langle s_{n-1}\rangle ={\gamma }_{n-1}(G)={\zeta }_1(G)$ lies in the centre of G. The first kernel equation $s_2^{-3}{s}_3^{-1}{s}_{n-1}^z=1$ is solvable by either $n=3$ , where $z=0$ , $s_3=1$ , $s_2^3=1$ , or $n=4$ , $z=1$ , where $s_2^3=1$ , $s_{n-1}^z=s_3$ . The second kernel equation $s_i^3=1$ is solvable by

Table 1. Shallow and deep TKT of 3-groups G with $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ .

either $i=n-1$ or $i=n-2$ . Thus, the deep transfer kernel is given by

$ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_1\mathrm{<mo\; stretchy="false">\; )\; </mo>}=(\begin{array}{l}{H}_1=\langle y\mathrm{,}{s}_2\rangle \simeq C_3\times C_3\text{if}n=3\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\text{extra}\text{special}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \\ H_1=\langle y\mathrm{,}{s}_2\mathrm{,}{s}_3\rangle \simeq C_3\times C_3\times C_3\text{if}n=\mathrm{4,}z=1\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\simeq {\text{Syl}}_3{A}_9\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \\ {\gamma }_{n-2}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; =\; </mo>}\langle s_{n-2}\mathrm{,}{s}_{n-1}\rangle \simeq C_3\times C_3\text{if}n=\mathrm{4,}z\ne 1\text{or}n\ge \mathrm{5,}a=\mathrm{0,}\hfill \\ {\gamma }_{n-2}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; /\; </mo>}{\gamma }_{n-1}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq \langle s_{n-2}\rangle \simeq C_3\text{if}n\ge \mathrm{5,}a=1\mathrm{<mo\; stretchy="false">\; (\; </mo>}{H}_1\text{non-abelian}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; .}\hfill \end{array}$ (3.4)

Second, we put $i=2$ . Then $H_2=\langle x,G^{\prime }\rangle$ and ${H^{\prime }}_2={\gamma }_3(G)=\langle s_3,\dots ,s_{n-1}\rangle$ . The outer transfer is determined by $T_2(x\cdot {H^{\prime }}_2)=x^3=s_{n-1}^w$ . The inner transfer is given by $T_2(s_j\cdot {H^{\prime }}_2)=s_j^{1+x+x^2}=s_j^3\cdot [s_j,x^3]\cdot [[s_j,x],x]=s_j^3{s}_{j+1}^3{s}_{j+2}=1$ , for all $j\ge 2$ , independently of $a\mathrm{,}n\mathrm{,}w\mathrm{,}z$ . Consequently, the deep transfer kernel is given by

$ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_2\mathrm{<mo\; stretchy="false">\; )\; </mo>}=(\begin{array}{l}{H}_2\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_2=\langle x\mathrm{,}{s}_2\mathrm{,}\dots \mathrm{,}{s}_{n-1}\rangle \mathrm{<mo>\; /\; </mo>}\langle s_3\mathrm{,}\dots \mathrm{,}{s}_{n-1}\rangle \simeq \langle x\mathrm{,}{s}_2\rangle \simeq C_3\times C_3\text{if}w=\mathrm{0,}\hfill \\ G^{\prime }\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_2=\langle s_2\mathrm{,}\dots \mathrm{,}{s}_{n-1}\rangle \mathrm{<mo>\; /\; </mo>}\langle s_3\mathrm{,}\dots \mathrm{,}{s}_{n-1}\rangle \simeq \langle s_2\rangle \simeq C_3\text{if}w=\pm 1.\hfill \end{array}$ (3.5)

Next, we put $i=3$ . Then $H_3=\langle xy,G^{\prime }\rangle$ and ${H^{\prime }}_3={\gamma }_3(G)=\langle s_3,\dots ,s_{n-1}\rangle$ . The outer transfer is determined by $T_3(xy\cdot {H^{\prime }}_3)={(xy)}^3=x^3{y}^3{s}_2^3{s}_3{s}_{n-1}^{-2a}=s_{n-1}^{w+z-2a}$ . For the inner transfer, we have $T_3(s_j\cdot {H^{\prime }}_3)=s_j^{1+xy+{(xy)}^2}=s_j^3\cdot [s_j,x{y}^3]\cdot [[s_j,xy],xy]=s_j^3{s}_{j+1}^3{s}_{j+2}=1$ , for all $j\ge 3$ , independently of $a\mathrm{,}n\mathrm{,}w\mathrm{,}z$ . The first kernel equation $s_{n-1}^{w+z-2a}=1$ $\iff$ $w+z-2a\equiv 0(\mathrm{mod}3)$ is solvable by either $a=w=z=0$ or $a=1$ , $w=-1$ .

Therefore, the deep transfer kernel is given by

$ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_3\mathrm{<mo\; stretchy="false">\; )\; </mo>}=(\begin{array}{l}{H}_3\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_3\simeq \langle xy\mathrm{,}{s}_2\rangle \simeq C_3\times C_3\text{if}\text{either}a=w=z=0\text{or}a=\mathrm{1,}w=-\mathrm{1,}\hfill \\ G^{\prime }\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_3\simeq \langle s_2\rangle \simeq C_3\text{otherwise}\mathrm{.}\hfill \end{array}$ (3.6)

Finally, we put i = 4. Then $H_4=\langle x{y}^2,G^{\prime }\rangle$ and ${H^{\prime }}_4={\gamma }_3(G)=\langle s_3,\dots ,s_{n-1}\rangle$ . The outer transfer is determined by $T_4(x{y}^2\cdot {H^{\prime }}_4)={(x{y}^2)}^3=x^3{y}^6{s}_2^6{s}_3^2{s}_{n-1}^{-2a}=s_{n-1}^{w+2z-2a}$ . The inner transfer is given by $T_4(s_j\cdot {H^{\prime }}_4)=s_j^{1+x{y}^2+{(x{y}^2)}^2}=s_j^3\cdot [s_j,x{y}^2{}^3]\cdot [[s_j,x{y}^2],x{y}^2]=s_j^3{s}_{j+1}^3{s}_{j+2}=1$ , for all $j\ge 3$ , independently of $a\mathrm{,}n\mathrm{,}w\mathrm{,}z$ . The first kernel equation $s_{n-1}^{w+2z-2a}\mathrm{<mo>\; =\; </mo>\; 1}$ $\iff$ $w+2z-2a\equiv 0(mod3)$ is solvable by either $a=w=z=0$ or $a=1$ , $w=-1$ .

Thus, the deep transfer kernel is given by

$ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_4)=(\begin{array}{l}{H}_4\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_4\simeq \langle x{y}^2\mathrm{,}{s}_2\rangle \simeq C_3\times C_3\text{if}\text{either}a=w=z=0\text{or}a=\mathrm{1,}w=-\mathrm{1,}\hfill \\ G^{\prime }\mathrm{<mo>\; /\; </mo>}{H^{\prime }}_4\simeq \langle s_2\rangle \simeq C_3\text{otherwise}\mathrm{.}\hfill \end{array}$

(3.7)

These finer results are summarized in terms of coarser cardinalities in Table 1.

□

4. Arithmetical Application to 3-Class Tower Groups

4.1. Real Quadratic Fields

As a final highlight of our progressive innovations, we come to a number theoretic application of Theorem 3.1, more precisely, the unambiguous identification of the pro-3 Galois group $G_3^{\infty }F=\text{Gal}\mathrm{<mo\; stretchy="false">\; (\; </mo>}{F}_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\infty \mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of the maximal unramified pro-3 extension $F_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\infty \mathrm{<mo\; stretchy="false">\; )\; </mo>}}$ , that is the Hilbert 3-class field tower, of certain real quadratic fields $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with fundamental discriminant $d>1$ , 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of elementary type $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}$ , and shallow transfer kernel type a.1, ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ , in its ground state with $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 9,9\; <mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ or in a higher excited state with $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3}}^e{\mathrm{,3}}^e{\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ , $e\ge 3$ .

The first field of this kind with $d=62501$ was discovered by Heider and Schmithals in 1982 [9]. They computed the sTKT ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ with four total 3-principalizations in the unramified cyclic cubic extensions $E_i\mathrm{<mo>\; /\; </mo>}F$ , $1\le i\le 4$ , on a CDC Cyber mainframe. The fact that $d=62501$ is a triadic irregular discriminant (in the sense of Gauss) with non-cyclic 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ has been pointed out earlier in 1936 by Pall [10] already. The second field of this kind with $d=152949$ was discovered by ourselves in 1991 by computing ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ on an AMDAHL mainframe [11]. In 2006, there followed $d=252977$ and $d=358285$ , and many other cases in 2009 [12] [13].

Generally, there are three contestants for the group $G=G_3^{\infty }F$ , for any assigned state $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3}}^e{\mathrm{,3}}^e{\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ , $e\ge 2$ , and the following Main Theorem admits their identification by means of the deep transfer kernel type (See their statistical distribution at the end of Section 4.1.).

Theorem 4.1. (3-class tower groups G of coclass cc(G) = 1 and type a.1.) Let $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ be a quadratic field with fundamental discriminant d, 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ , and shallow transfer kernel type a.1, ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ .

Then F is real with $d>1$ , the 3-class tower group $G=G_3^{\infty }F$ of F has coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ , and the relational parameters $n\ge 5$ and $w\in \mathrm{<mo>\; \{\; </mo>}-\mathrm{1,0,1\; <mo>\; \}\; </mo>}$ of $G\simeq G_1^n\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ are given in dependence on the deep transfer kernel type ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ as follows:

$\begin{array}{lllll}G\simeq G_1^{\mathrm{2\; <mo\; stretchy="false">\; (\; </mo>}e+\mathrm{1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0\; <mo\; stretchy="false">\; )\; </mo>}\hfill & \text{with}\hfill & n=2(e+1),w=0\hfill & \iff \hfill & {\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,9,3,3\; <mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \\ G\simeq G_1^{\mathrm{2\; <mo\; stretchy="false">\; (\; </mo>}e+\mathrm{1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,}-\mathrm{1\; <mo\; stretchy="false">\; )\; </mo>}\hfill & \text{with}\hfill & n=2(e+1),w=-1\hfill & \iff \hfill & {\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3,9,9\; <mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \\ G\simeq G_1^{\mathrm{2\; <mo\; stretchy="false">\; (\; </mo>}e+\mathrm{1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,1\; <mo\; stretchy="false">\; )\; </mo>}\hfill & \text{with}\hfill & n=2(e+1),w=1\hfill & \iff \hfill & {\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3,3,3\; <mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \end{array}$ (4.1)

where we suppose that the state of type a.1 is determined by the transfer target type $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3}}^e{\mathrm{,3}}^e{\mathrm{<mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ with $e\ge 2$ .

Proof. Let $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ be a quadratic field with 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ , denote by $E_1\mathrm{,}\dots \mathrm{,}{E}_4$ its four unramified cyclic cubic extensions and by $T_{E_i\mathrm{<mo>\; /\; </mo>}F}\mathrm{<mo>\; :\; </mo>}{\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\to {\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1}\le i\le \mathrm{4\; <mo\; stretchy="false">\; )\; </mo>}$ the transfer homomorphisms of 3-classes.

If the 3-principalization is total, that is $ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{E_i\mathrm{<mo>\; /\; </mo>}F}\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ , for each $1\le i\le 4$ , then F must be a real quadratic field with positive fundamental discriminant $d>1$ , since the order of the principalization kernels $ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{E_i\mathrm{<mo>\; /\; </mo>}F}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of an imaginary quadratic field F is bounded from above by $(U_F:N_{E_i/F}{U}_{E_i})\cdot E_i:F=1\cdot 3=3$ , according to the Theorem on the Herbrand quotient of the unit groups $U_{E_i}$ .

By the Artin reciprocity law of class field theory [1] [14], the principalization type $\varkappa (F)=(0000)$ of the field F corresponds to the shallow transfer kernel type ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}G)=(0000)$ of the 3-class tower group $G=\text{Gal}\mathrm{<mo\; stretchy="false">\; (\; </mo>}{F}_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\infty \mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of F, and the abelian type invariants ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq 1^2$ of the 3-class group of F correspond to the abelian quotient invariants $G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }\simeq 1^2$ of G.

According to [2], a finite 3-group G with $G\mathrm{<mo>\; /\; </mo>}{G}^{\prime }\simeq 1^2$ and ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}G)=(0000)$ must be of coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ . Table 1 shows that either $G\simeq G_0^n\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0\; <mo\; stretchy="false">\; )\; </mo>}$ of type a.1^* with $n\ge 2$ or $G\simeq G_1^n\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of type a.1 with $n\ge 5$ and $-1\le w\le 1$ .

For a real quadratic field F, the relation rank $d_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=di{m}_{{\mathbb{F}}_3}{H}_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{,}{\mathbb{F}}_3)$ of the 3-class tower group $G=G_3^{(\infty )}F$ is bounded by $d_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}\le 3$ [[15], Thm. 1.3, pp. 75-76]. Consequently, G cannot be a non-abelian mainline vertex $G_0^n\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0\; <mo\; stretchy="false">\; )\; </mo>}$ with $n\ge 3$ of the coclass-1 tree ${\mathcal{T}}^1\mathrm{<mo\; stretchy="false">\; (\; </mo>}R\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with root $R=C_3\times C_3$ , since all these vertices have the relation rank 4. According to [[12], Thm. 4.1 (1), p. 486], G cannot be the abelian root $R=G_0^2(0,0)$ either, and we must have $G\simeq G_1^n\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,}w\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with $n\ge 5$ and $w\in \mathrm{<mo>\; \{\; </mo>}-\mathrm{1,0,1\; <mo>\; \}\; </mo>}$ .

Now the claim is a consequence of Theorem 3.1 and Table 1. □

Table 2 shows that the ground state $\tau (F)=(9,9),{(3,3)}^3$ of the sTKT ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F)=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ has the nice property that the smallest three discriminants already realize three different 3-class tower groups $G=G_3^{\infty }F\simeq \langle 729,i\rangle$ with $i\in \mathrm{<mo>\; \{\; </mo>\; 99,100,101\; <mo>\; \}\; </mo>}$ , identified by their dTKT ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ .

In Table 3, we see that the first excited state $\tau (F)=(27,27),{(3,3)}^3$ of the sTKT ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ does not behave so well: although the smallest two discriminants [12] [13] [16] [17] already realize two different 3-class tower groups $G=G_3^{\infty }F\simeq \langle 6561,i\rangle$ with $i\in \mathrm{<mo>\; \{\; </mo>\; 2225,2227\; <mo>\; \}\; </mo>}$ , we have to wait for the seventh occurrence until $\langle \mathrm{6561,2226}\rangle$ is realized, as the dTKT ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ shows. The counter 7 is a typical example of a statistic delay.

The second excited state $\tau (F)=(81,81),{(3,3)}^3$ of the sTKT

Table 2. Deep TKT of 3-class tower groups G with $\tau (G)=(9,9),{(3,3)}^3$ .

Table 3. Deep TKT of 3-class tower groups G with $\tau (G)=(27,27),{(3,3)}^3$ .

${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ , however, is well-behaved again: the smallest three discriminants already realize three different 3-class tower groups $G=G_3^{\infty }F\simeq G_1^{10}(0,w)$ with $w\in \mathrm{<mo>\; \{\; </mo>\; 0,}-\mathrm{1,1\; <mo>\; \}\; </mo>}$ , identified by their dTKT ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ . (For logarithmic orders $\ge 9$ , no SmallGroup identifiers exist.) See Table 4.

In all tables, the shortcut MD means the minimal discriminant [[7], Dfn. 6.2, p. 148].

The diagram in Figure 1 visualizes the initial eight branches of the coclass tree ${\mathcal{T}}^1\mathrm{<mo\; stretchy="false">\; (\; </mo>}R\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with abelian root $R=\langle 9,2\rangle \simeq C_3\times C_3$ . Basic definitions, facts, and notation concerning general descendant trees of finite p-groups are summarized briefly in [[18], §2, pp. 410-411] [19]. They are discussed thoroughly in the broadest detail in the initial sections of [20]. Descendant trees are crucial for recent progress in the theory of p-class field towers [15] [21] [22], in particular for describing the mutual location of the second p-class group ${\text{G}}_p^{2}F$ and the p-class tower group ${\text{G}}_p^{\infty }F$ of a number field G. Generally, the vertices of the coclass tree in the figure represent isomorphism classes of finite 3-groups. Two vertices are connected by a directed edge $G\to H$ if H is isomorphic to the last lower central quotient $G\mathrm{<mo>\; /\; </mo>}{\gamma }_c\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ , where $c=\text{cl}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=n-1$ denotes the nilpotency class of G, and $\left|G\right|=3\left|H\right|$ , that is, ${\gamma }_c\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3$ is cyclic of order 3. See also [[18], §2.2, p. 410-411] and [[20], §4, p. 163-164].

The vertices of the tree diagram in Figure 1 are classified by using various symbols:

1) big contour squares , represent abelian groups,

2) big full discs ・ represent metabelian groups with at least one abelian maximal subgroup,

3) small full discs ・ represent metabelian groups without abelian maximal subgroups.

The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library [23] [24] of MAGMA [25]. We omit the orders, which are given on the left hand scale. The sTKT ${\varkappa }_s$ [[2] Thm. 2.5, Tbl. 6-7], in the bottom rectangle concerns all vertices located vertically above. The first component $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1\; <mo\; stretchy="false">\; )\; </mo>}$ of the TTT [[26] [27], Dfn. 3.3, p. 288] in the left rectangle concerns vertices G on the same horizontal level containing an abelian maximal subgroup. It is given in logarithmic notation. The periodicity with length 2 of branches, $\mathcal{B}\mathrm{<mo\; stretchy="false">\; (\; </mo>}j\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq \mathcal{B}\mathrm{<mo\; stretchy="false">\; (\; </mo>}j+\mathrm{2\; <mo\; stretchy="false">\; )\; </mo>}$ for $j\ge 4$ , sets in with branch $\mathcal{B}\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 4\; <mo\; stretchy="false">\; )\; </mo>}$ , having a root of order 3⁴.

3-class tower groups $G=G_3^{\infty }F$ with coclass $\text{cc}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}=1$ of real quadratic

Table 4. Deep TKT of 3-class tower groups G with $\tau (G)=(81,81),{(3,3)}^3$ .

Figure 1. Distribution of minimal discriminants for $G_3^{\infty }F$ on the coclass-1 tree ${\mathcal{T}}^1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\langle \mathrm{9,2}\rangle \mathrm{<mo\; stretchy="false">\; )\; </mo>}$

fields $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ are located as arithmetically realized vertices on the tree diagram in Figure 1. The minimal fundamental discriminants d, i.e. the MDs, are indicated by underlined boldface integers adjacent to the oval surrounding the realized vertex [6] [24] [25].

The double contour rectangle surrounds the vertices which became distinguishable by the progressive innovations in the present paper and were inseparable up to now.

In Table 5, we give the isomorphism type of the 3-class tower group $G=G_3^{\infty }F$ of all real quadratic fields $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ with 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ and shallow transfer kernel type a.1, ${\varkappa }_s=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ , in its ground state $\tau (F)=(9,9),{(3,3)}^3$ , for the complete range $1<d<{10}^7$ of 150 fundamental discriminants d. It was determined by means of Theorem 4.1, applied to the results of computing the (restricted) deep transfer kernel type ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}={\mathrm{<mo\; stretchy="false">\; (\; </mo>\; <mo>\; \#\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{F_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}{E}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}_{2\le i\le 4}$ , consisting of the orders of the 3-principalization kernels of those unramified cyclic cubic extensions $E_i$ , $2\le i\le 4$ , in the Hilbert 3-class field $F_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1\; <mo\; stretchy="false">\; )\; </mo>}}$ of F whose 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ is of type $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}$ . These trailing three components of the TTT $\tau (F)=(9,9),{(3,3)}^3$ were called its stable part in [[6], Dfn. 5.5, p. 84]. The computations were done with the aid of the computational algebra system MAGMA [25]. The 3-principalization kernel of the remaining extension $E_1$ with 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_1\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of type $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 9,9\; <mo\; stretchy="false">\; )\; </mo>}$ does not contain essential information and can be omitted. This leading component of the TTT $\tau (F)=(9,9),{(3,3)}^3$ was called its polarized part in [[6], Dfn. 5.5, p. 84]. For more details on the concepts stabilization and polarization, see [[6], §6, pp. 90-95].

A systematic statistical evaluation of Table 5 shows that, with respect to the complete range $1<d<{10}^7$ , the group $G\simeq \langle \mathrm{729,99}\rangle$ occurs most often with a clearly elevated relative frequency of 44%, whereas $G\simeq \langle \mathrm{729,100}\rangle$ and $G\simeq \langle \mathrm{729,101}\rangle$ share the common lower percentage of 28%, although the automorphism group $\text{Aut}\mathrm{<mo\; stretchy="false">\; (\; </mo>}G\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of all three groups has the same order. However, the proportion $\mathrm{44\; <mo>\; :\; </mo>\; 28\; <mo>\; :\; </mo>\; 28}$ for the upper bound 10⁷ is obviously not settled yet, because there are remarkable fluctuations, as Table 6 shows. According to Boston, Bush and Hajir [28] [29], we have to expect an asymptotic limit $\mathrm{33\; <mo>\; :\; </mo>\; 33\; <mo>\; :\; </mo>\; 33}$ of the proportions for $d\to \infty$ .

4.2. Totally Real Dihedral Fields

In fact, we have computed much more information with MAGMA than mentioned at the end of the previous Section 4.1. To understand the actual scope of our numerical results it is necessary to recall that each unramified cyclic cubic relative extension $E_i\mathrm{<mo>\; /\; </mo>}F$ , $1\le i\le 4$ , gives rise to a dihedral absolute extension $E_i\mathrm{<mo>\; /\; </mo>}\mathbb{Q}$ of degree 6, that is an $S_3$ -extension [[12], Prp. 4.1, p. 482]. For the trailing three fields $E_i$ , $2\le i\le 4$ , in the stable part of the TTT $\tau (F)=(9,9),{(3,3)}^3$ , i.e. with ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of type $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}$ , we have constructed the unramified cyclic cubic extensions ${\tilde{E}}_{i\mathrm{,}j}\mathrm{<mo>\; /\; </mo>}{E}_i$ , $1\le j\le 4$ , and determined the Artin pattern $\text{AP}\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of

Table 5. Statistics of 3-class tower groups G with $\tau (G)=(9,9),{(3,3)}^3$ .

Table 6. Proportions of 3-class tower groups $G\simeq \langle \mathrm{729,}i\rangle$ with $i\in \mathrm{<mo>\; \{\; </mo>\; 99,100,101\; <mo>\; \}\; </mo>}$ .

$E_i$ , in particular, the 3-principalization type of $E_i$ in the fields ${\tilde{E}}_{i\mathrm{,}j}$ . The dihedral fields $E_i$ of degree 6 share a common polarization ${\tilde{E}}_{i,1}=F_3^{(1)}$ , the Hilbert 3-class field of F, which is contained in the relative 3-genus field ${\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo>\; /\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}^{\ast }$ , whereas the other extensions ${\tilde{E}}_{i\mathrm{,}j}$ with $2\le j\le 4$ are non-abelian over F, for each $2\le i\le 4$ . Our computational results suggest the following conjecture concerning the infinite family of totally real dihedral fields $E_i$ for varying real quadratic fields F.

Conjecture 4.1. (3-class tower groups $\mathcal{G}$ of totally real dihedral fields.) Let $F=\mathbb{Q}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\sqrt{d}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ be a real quadratic field with fundamental discriminant $d>1$ , 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ , and shallow transfer kernel type a.1, ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>}$ , in the ground state with transfer target type $\tau \mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}~{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 9,9\; <mo\; stretchy="false">\; )\; </mo>\; ,\; <mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}}^3$ . Let $E_2\mathrm{,}{E}_3\mathrm{,}{E}_4$ be the three unramified cyclic cubic relative extensions of F with 3-class group ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of type $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3\; <mo\; stretchy="false">\; )\; </mo>}$ .

Then $E_i\mathrm{<mo>\; /\; </mo>}\mathbb{Q}$ is a totally real dihedral extension of degree 6, for each $2\le i\le 4$ , and the connection between the component ${\varkappa }_d{\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i=\mathrm{<mo>\; \#\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{F_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}{E}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of the deep transfer kernel type ${\varkappa }_d\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of F and the 3-class tower group ${\mathcal{G}}_i=G_3^{\infty }{E}_i=\text{Gal}\mathrm{<mo\; stretchy="false">\; (\; </mo>\; <mo\; stretchy="false">\; (\; </mo>}{E}_i{\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\infty \mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ of $E_i$ is given in the following way:

$\begin{array}{llll}{\varkappa }_d{\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i=3\hfill & \iff \hfill & {\mathcal{G}}_i\simeq \langle \mathrm{243,27}\rangle \hfill & \text{with}{\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\mathcal{G}}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1,0,0,0\; <mo\; stretchy="false">\; )\; </mo>\; ,}\hfill \\ {\varkappa }_d{\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i=9\hfill & \iff \hfill & {\mathcal{G}}_i\simeq \langle \mathrm{243,26}\rangle \hfill & \text{with}{\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\mathcal{G}}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 0,0,0,0\; <mo\; stretchy="false">\; )\; </mo>\; .}\hfill \end{array}$ (4.2)

Remark 4.1. The conjecture is supported by all $3\cdot 150=450$ totally real dihedral fields $E_i$ which were involved in the computation of Table 5. A provable argument for the truth of the conjecture is the fact that ${\varkappa }_d{\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i\mathrm{<mo>\; =\; </mo>\; <mo>\; \#\; </mo>}ker\mathrm{<mo\; stretchy="false">\; (\; </mo>}{T}_{F_3^{\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 1\; <mo\; stretchy="false">\; )\; </mo>}}\mathrm{<mo>\; /\; </mo>}{E}_i}\mathrm{<mo\; stretchy="false">\; )\; </mo>}=\mathrm{<mo>\; \#\; </mo>}{\varkappa }_s{\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_1\mathrm{<mo>\; =\; </mo>\; <mo>\; \#\; </mo>}{\varkappa }_s{\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\mathcal{G}}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_1$ , for $2\le i\le 4$ , but it does not explain why the sTKT ${\varkappa }_s\mathrm{<mo\; stretchy="false">\; (\; </mo>}{\mathcal{G}}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}$ is a.2 with a fixed point if ${\varkappa }_d{\mathrm{<mo\; stretchy="false">\; (\; </mo>}F\mathrm{<mo\; stretchy="false">\; )\; </mo>}}_i=3$ . It is interesting that a dihedral field $E_i$ of degree 6 is satisfied with a non-s group, such as $\langle \mathrm{243,27}\rangle$ , as its 3-class tower group. On the other hand, it is not surprising that a mainline group, such as $\langle \mathrm{243,26}\rangle$ with sTKT a.1* and relation rank $d_2=4$ , is possible as ${\mathcal{G}}_i=G_3^{\infty }{E}_i$ , since the upper Shafarevich bound for the relation rank of the 3-class tower group of a totally real dihedral field $E_i$ of degree 6 with ${\text{Cl}}_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}{E}_i\mathrm{<mo\; stretchy="false">\; )\; </mo>}\simeq C_3\times C_3$ is given by $\rho +r_1+r_2-1=2+6+0-1=7>4$ [[15], Thm. 1.3, p. 75].

Assuming an asymptotic limit $\mathrm{33\; <mo>\; :\; </mo>\; 33\; <mo>\; :\; </mo>\; 33}$ of the proportion of the real quadratic 3-class tower groups $G\in \mathrm{<mo>\; \{\; </mo>}\langle \mathrm{729,99}\rangle \mathrm{,}\langle \mathrm{729,100}\rangle \mathrm{,}\langle \mathrm{729,101}\rangle \mathrm{<mo>\; \}\; </mo>}$ for the ground state of sTKT a.1, we can also conjecture an asymptotic limit $\mathrm{33\; <mo>\; :\; </mo>\; 66}$ of the corresponding totally real dihedral 3-class tower groups ${\mathcal{G}}_i\in \mathrm{<mo>\; \{\; </mo>}\langle \mathrm{243,26}\rangle \mathrm{,\; <mo>\; \{\; </mo>}\langle \mathrm{243,27}\rangle \mathrm{<mo>\; \}\; </mo>}$ , since the restricted dTKTs $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 9,3,3\; <mo\; stretchy="false">\; )\; </mo>}$ , $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,9,9\; <mo\; stretchy="false">\; )\; </mo>}$ , $\mathrm{<mo\; stretchy="false">\; (\; </mo>\; 3,3,3\; <mo\; stretchy="false">\; )\; </mo>}$ together contain three times the 9 and six times the 3 in Equation (4.2).

Acknowledgements

The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): P 26008-N25. Note added in proof: While this paper was under review, we succeeded in proving Conjecture 4.1with the aid of Theorems 5.1, 6.1, 6.5,on pages 676, 678, 682 in [30].

Conflicts of Interest

The authors declare no conflicts of interest.

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