Open access peer-reviewed chapter

# Modeling Rooted in‐Trees by Finite p‐Groups

Written By

Daniel C. Mayer

Submitted: October 18th, 2016 Reviewed: March 22nd, 2017 Published: December 20th, 2017

DOI: 10.5772/intechopen.68703

From the Edited Volume

## Graph Theory

Edited by Beril Sirmacek

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## Abstract

Graph theoretic foundations for a kind of infinite rooted in-trees T(R)=(V,E) with root R, weighted vertices v ∈ V, and weighted directed edges e∈E⊂V×V are described. Vertex degrees deg(v) are always finite but the trees contain infinite paths (vi)i≥0. A concrete group theoretic model of the rooted in-trees T(R) is introduced by representing vertices by isomorphism classes of finite p-groups G, for a fixed prime p, and directed edges by epimorphisms π: G → πG of finite p-groups with characteristic kernels ker(π). The weight of a vertex G is realized by its nuclear rank n(G) and the weight of a directed edge π is realized by its step size s(π)=logp(#ker(π)). These invariants are essential for understanding the phenomenon of multifurcation. Pattern recognition methods are used for finding finite subgraphs which repeat indefinitely. Several periodicities admit the reduction of the complete infinite graph to finite patterns. The proof is based on infinite limit groups and successive group extensions. It is underpinned by several explicit algorithms. As a final application, it is shown that fork topologies, arising from repeated multifurcations, provide a convenient description of complex navigation paths through the trees, which are of the greatest importance for recent progress in determining p-class field towers of algebraic number fields.

### Keywords

• rooted directed in‐trees
• descendant trees
• infinite paths
• vertex distance
• weighted edges
• pattern recognition methods
• pattern classification
• independent component analysis
• graph dissection
• finite p‐groups
• projective limits
• periodicity
• group extensions
• nuclear rank
• multifurcation
• presentations
• commutators
• central series

## 1. Introduction

In Section 2, we describe the abstract graph theoretic foundations for a kind of infinite rooted in‐trees T(R)=(V,E)with root R, weighted vertices vV, and weighted directed edges eEV×V, which are suited perfectly for describing the crucial phenomenon of multifurcationin Section 2.3. The vertex degrees deg(v)are always finite, but the trees contain infinite paths (vi)i0. In Section 3, we introduce a group theoretic model of the rooted in‐trees T(R). Vertices are represented by isomorphism classes of finite p‐groups G, for a fixed prime number p. Directed edges are represented by epimorphisms π:GπGof finite p‐groups with characteristic kernels ker(π). The weight of a vertex Gis realized by its nuclear rankn(G), and the weight of a directed edge πis realized by its step sizes(π)=logp(#ker(π)). Since the structure of our rooted in‐trees is rather complex, we use pattern recognitionmethods in Section 3.1 for finding finite subgraphs which repeat indefinitely as branches of coclass subtrees, thus giving rise to a first periodicity. Additionally, we employ independent component analysisfor obtaining a graph dissection into pruned subtrees, either by Galois action in Section 3.2.1 or by Artin transfers in Section 3.2.2. A second periodicityof pruned coclass subtrees eventually admits the reduction of the complete infinite graph T(R)to finite patterns. Evidence of these newly discovered periodic bifurcationsis provided by a mixture of bottom up techniques, using successive extensions by means of p‐covering groups in the pgroup generation algorithmand top down techniques using infinite limitsand their finite quotientsin Sections 3.4 and 3.5. As a coronation of this chapter, we show in Sections 3.6 and 4 that fork topologiesprovide a convenient description of very complex navigation paths through the trees, arising from repeated multifurcations, which are of the greatest importance for recent progress in determining pclass field towersFp()of algebraic number fields F.

## 2. Underlying abstract graph theory

Let G=(V,E)be a graphwith set of verticesVand set of edgesE. We expressly admit infinite sets Vand E, but we assume that the in and out degree of each vertex is finite.

### 2.1. Directed edges and paths

In this chapter, we shall be concerned with directed graphs(digraphs) whose edges are rather ordered pairs (v1,v2)V×Vthan only subsets {v1,v2}Vwith two elements. Such a directed edgee=(v1,v2)is also denoted by an arrow v1v2with starting vertex v1and ending vertex v2. Thus, we have EV×V. Now, infinitude comes in.

Definition 2.1.(Finite and infinite paths.)

A finite pathof length0in Gis a finite sequence (vi)0iof vertices viVsuch that (vi,vi+1)Efor 0i1. We call v0, respectively, v, the starting vertex, respectively, ending vertex, of the path. The degenerate case of a single vertex (v0)is called a point pathof length =0.

An infinite pathin Gis an infinite sequence (vi)i0of vertices viVsuch that (vi+1,vi)Efor all i0. In this case, v0is the ending vertex of the path, and there is no starting vertex.

### 2.2. Rooted in‐trees with parent operator

Our attention will even be restricted to rooted in‐treesG=T(R), that is, connected digraphs without cycles such that the root vertexRhas out‐degree 0, whereas any other vertex vV\{R}has out‐degree 1. A vertex with in‐degree at least 1is called capable, whereas a vertex with in‐degree 0is called a leaf. For a rooted in‐tree, we can define the parent operator as follows.

Definition 2.2.Let T(R)=(V, E)be a rooted in‐tree. Then, the mapping π: V\{R}V, vπv, where (v,πv)Eis the unique edge with starting vertex v, is called the parent operatorof T(R). For each vertex vV, there exists a unique finite root pathfrom vto the root R,

v=π0vπ1vπ2vπ1vπv=R,

expressed by iterated applications of the parent operator and with some length0. Each vertex in the root path of vis called an ancestorof v.

The descendant treeT(a)=(V(a),E(a))of a vertex aVis the subtree of T(R)=(V,E)consisting of vertices vwith ancestor a, that is vV(a):={uV|(j0) πju=a}, and edges eE(a):=E(V(a)×V(a)).

A vertex uVis called an immediate descendant(or child) of a vertex aV, if there exists a directed edge (u,a)E. In this case, a=πuis necessarily the parentof u.

We can define a partial orderon the vertices u,aVof the tree T(R)by putting uaif uT(a), that is, if uis descendant of aand ais ancestor of u. The root Ris the minimum.

The root Ris always a common ancestor of two vertices u,vV. By the forkof uand v, we understand their biggest common ancestor, denoted by Fork(u,v), which admits a measure.

Definition 2.3.(Vertex distance.) The sum u+vof the path lengths from two vertices u,vVto their fork is called the distanced(u,v)of the vertices.

### 2.3. Mainlines and multifurcation

We shall also need weight functionswith nonnegative integer values for vertices wV: VN0, and with positive integer values for edges wE: EN. In particular, the sets of vertices and edges have disjoint partitions

V=n0Vn with Vn:={vV|wV(v)=n} for n0,E=s1Es with Es:={eE|wE(e)=s} for s1,E1

such that V0is precisely the set of leavesof the tree T(R). Thus, there arise weighted measures.

Definition 2.4.(Path weight and weighted distance.)

By the path weightof a finite path (vi)0iwith length 0in T(R)such that (vi,vi+1)Esifor 0i1, we understand the sum i=01si. The sum wu+wvof the path weights from two vertices u,vVto their fork is called the weighted distancew(u,v)of the vertices.

In Definitions 2.5 and 2.6, some concepts are introduced using the minimal possible weight.

Definition 2.5.(Mainlines and minimal trees.) An infinite path (vi)i0in T(R)with edges of weight 1, that is, such that (vi+1,vi)E1for all i0, is called a mainlinein T(R).

The minimal treeT1(a)=(V1(a),E1(a))of a vertex aVis the subtree of the descendant tree T(a)=(V(a),E(a))consisting of vertices v, whose root path in T(a)possesses edges eof weight 1only, that is vV1(a):={uV(a)|(0j<) (πju,πj+1u)E1}, and edges eE1(a):=E(a)(V1(a)×V1(a)).

Definition 2.6.(Branches.) Let (vi)i0be a mainline in T(R). For i0, the difference set B(vi):=T1(vi)\T1(vi+1)of minimal trees is called the branchwith root viof the minimal tree T1(v0). The branches give rise to a disjoint partition T1(v0)=˙i0B(vi).

Finally, we complete our abstract graph theoretic language by considering arbitrary weights.

Definition 2.7.(Multifurcation.)

Let n2be a positive integer. A vertex aVnhas an n‐fold multifurcationif its in‐degree is an n‐fold sum N1+N2++Nndue to Ns1incoming edges of weight s, for each 1sn. That is, we define counters Nsof all incoming edges of weight s, and additionally, we have counters Csof all incoming edges of weight swith capable starting vertex

Ns:=Ns(a):=#{eEs|e=(u,a) for some uV},Cs:=Cs(a):=#{eEs|e=(u,a) for some uV with wV(u)1}.E2

We also define an ordering and a notation [1] for immediate descendants of aby writing a#s;ifor the ith immediate descendant with edge of weight s, where 1snand 1iNs.

## 3. Concrete model in p‐group theory

Now, we introduce a group theoretic model of the rooted in‐trees T(R)=(V,E)in Section 2. Vertices vVare represented by isomorphism classes of finite p‐groups G, for a fixed prime number p. Directed edges eEare represented by epimorphisms π:GπGof finite p‐groups with characteristic kernels ker(π)=γcG, where c:=cl(G)denotes the nilpotency class of Gand (γiG)i1is the lower central series of G.

We emphasize that the symbol πis used now intentionally for two distinct mappings, the abstract parent operator π: V\{R}V, vπv, in Definition 2.2, and the concrete natural projection onto the quotient π:GπGG/γcG, gπ(g)=gγcG, for each individual vertex G=vV\{R}, which should precisely be denoted by π=πG, but we omit the subscript, since there is no danger of misinterpretation. In both views, πGis the parent of G.

The weight of a vertex Gis realized by its nuclear rankn(G)([2], section 14, eqn. (28), p. 178) and the weight of a directed edge π:GπGis realized by its step sizes(π)=logp(#γcG)([2], section 17, eqn. (33), p. 179). These invariants are essential for understanding the phenomenon of multifurcationin Definition 2.7. In particular, we can hide multifurcation by restricting all edges πto step size s(π)=1, that is, by considering the minimal tree T1(v)instead of the entire descendant tree T(v)of a vertex vV. In our concrete p‐group theoretic model, all vertices Gof a minimal tree share a common coclass, which is the additive complement cc(G):=lo(G)cl(G)of the (nilpotency) class c=cl(G)with respect to the logarithmic orderlo(G):=logp(ord(G))of G. Generally, the logarithmic order of an immediate descendant Gwith parent πGincreases by the step size, lo(G)=lo(πG)+s(π), since logp(#πG)=logp(#(G/kerπ))=logp(#G)logp(#kerπ). Consequently, the coclass remains fixed in a minimal tree with s(π)=1, since

cc(G)=lo(G)cl(G)=lo(πG)+1(cl(πG)+1)=lo(πG)cl(πG)=cc(πG).

A minimal tree T1(G)which contains a unique infinite mainline is called a coclass tree. It is denoted by T(r)(G):=T1(G)when its root Gis of coclass r:=cc(G). For further details, see ([2], section 5, p. 164).

In view of the principal goals of this chapter, we must specify our intended situation even more concretely. We put p:=3, the smallest odd prime number, and we select as the root either R:=243,6or R:=243,8, characterized by its SmallGroup identifier [3]. These are metabelian 3‐groups of order #R=243=35, logarithmic order lo(R)=5, class c=3, and coclass r=2.

### 3.1. Periodicity of finite patterns

Within the frame of the above‐mentioned model with p=3for the theory of rooted in‐trees as developed in Section 2, the following finiteness and periodicity statement becomes provable.

The virtual periodicityof depth‐pruned branches of coclass trees has been proven rigorously with analytic methods (using zeta functions and cone integrals) by du Sautoy [4] in 2000, and with algebraic methods (using cohomology groups) by Eick and Leedham‐Green [5] in 2008. We recall that a coclass tree contains a unique infinite path of edges πwith uniform step size s(π)=1, the so‐called mainline. Pattern recognition and pattern classification concern the branches.

Theorem 3.1.(A finite periodically repeating pattern.)

Among the vertices of any mainline(vi)i0inT(R), there exists aperiodic rootvϱwithϱ0and aperiod lengthλ1such that the branches

B(vi+λ)B(vi)

are isomorphicfinitegraphs, for alliϱ. Up to a finite preperiodic component, the minimal treeT1(v0)consists of periodically repeating copies of the finite pattern˙i=0λ1B(vϱ+i).

Proof.According to [4, 5], the claims are true for pruned branches with any fixed depth. However, for p=3and under the pruning operation on T(R)described in Section 3.2.2, the virtual periodicity becomes a strict periodicity, since the depth is bounded uniformly for all branches.□

Before we visualize a particular instance of Theorem 3.1 in the diagram of Figure 1, we have to establish techniques for disentangling dense branches of high complexity.

### 3.2. Graph dissection by independent component analysis

#### 3.2.1. Dissection by Galois action

Figure 1 visualizes a graph dissection of the tree T(R)by independent component analysis. This technique drastically reduces the complexity of visual representations and avoids overlaps of dense subgraphs. The left hand scale gives the order of groups whose isomorphism classes are represented by vertices of the graph. The mainline skeleton (black) connects branches of non σ‐groups (red) in the left subfigure and branches of σ‐groups (green) in the right subfigure. This terminology has its origin in the action of the Galois group Gal(F/Q)on the abelianization M/M, when a vertex of T(R)is realized as second 3‐class group M:=Gal(F3(2)/F)of an algebraic number field F. For quadratic fields F, we obtain σ‐groups.

Definition 3.1.A σgroupGadmits an automorphism σAut(G)acting as inversion σ(x)=x1on the commutator quotient G/G.

The actual graph T(R)consists of the overlay (superposition) of both subfigures in Figure 1. Infinite mainlines are indicated by arrows. The periodic bifurcations form an infinite path with edges of alternating step sizes 1and 2, according to Theorem 3.2. We call it the maintrunk.

With the aid of Figure 1, a particular instance of Theorem 3.1 can be expressed in a more concrete and ostensive way by taking the tree root as the ending vertex v0:=Rof the mainline (vi)i0, and by using the variable class c3and the fixed coclass r=2as parameters describing all mainline vertices Mc(r):=vc3. The periodic root is M5(2)=v2with ϱ=2and the period length is λ=2. The finite periodic pattern consists of the two branches B(M5(2))={M5(2),G6,1(2),G6,2(2)}(red) and B(M6(2))={M6(2),G7,1(2),G7,2(2),G7,3(2)}(green). The preperiod is irregular and consists of the two branches B(M3(2))={M3(2),G4,1(2),G4,2(2),T5,1(2),T5,2(2)}(red) and B(M4(2))={M4(2),G5,1(2),G5,2(2),G5,3(2)}(green). But M4(2)is not coclass‐settled, has nuclear rank n=2and gives rise to a bifurcation with immediate descendants S5,1(3),S5,2(3),S5,3(3),M5(3)(green) of step size s=2.

#### 3.2.2. Dissection by Artin transfers

In Figure 1, we have tacitly used a second technique of graph dissection by independent component analysis. Figure 2 is restricted to the coclass tree T(2)(R)with exemplary root R=243,8, which is the leftmost coclass tree in both subfigures of Figure 1. However, now this coclass tree is drawn completely up to logarithmic order 15, containing both, nonσ‐branches and σ‐branches. The tree is embedded in a kind of coordinate system having the transfer kernel type (TKT) ϰas its horizontal axis and the first component τ(1)of the transfer target type (TTT) τas its vertical axis ([6], Dfn. 4.2, p. 27). It is convenient to employ a second graph dissection, according to three fundamental types of transfer kernels

• the vertices with simpletypes E.8, ϰ=(1231), and E.9, ϰ=(2231), which are leaves (left of the mainline), except those of order 36,

• the vertices with scaffold(or skeleton) type c.21, ϰ=(0231), which are either infinitely capable mainline vertices or nonmetabelian leaves (immediately right of the mainline),

• the vertices with complextype G.16, ϰ=(4231), which are capable at depth 1and give rise to a complicated brushwood of various descendants (right of the mainline).

The tacit omission in Figure 1 concerns all vertices with complex type and the leaves with scaffold type. Our main results in this chapter will shed complete light on all mainline vertices and the vertices with simple types. Underlined boldface integers in Figure 2 indicate the minimal discriminants dof (real and imaginary) quadratic fields F=Q(d)whose second 3‐class group G3(2)F:=Gal(F3(2)/F)realizes the vertex surrounded by the adjacent oval. Three leaves of type E.8are drawn with red color, because they will be referred to in Theorem 4.1 on 3‐class towers.

### 3.3. Periodicity of infinite patterns

With the aid of a combination of top down and bottom up techniques, we are now going to provide evidence of a new kind of periodic bifurcationsin pruned descendant trees which contain a unique infinite path of edges πwith strictly alternating step sizes s(π)=1and s(π)=2, the so‐called maintrunk. It is very important that the trees are prunedin the sense explained at the end of the preceding Section 3.2.2; for otherwise, the maintrunk will not be unique. In fact, each of our pruned descendant trees T(R)is a countable disjoint union of pruned coclass trees T(r), r2, which are isomorphic as infinite graphs and connected by edges of weight 2, and finite batches T0(r), r3, of sporadic vertices outside of coclass trees. The top down and bottom up techniques are implemented simultaneously in two recursive Algorithms 3.1 and 3.2.

The first Algorithm 3.1 recursively constructs the mainline vertices Mc(r), with class c2r1, of the coclass tree T(r)T(R), for an assigned value r2, by means of the bottom uptechnique. In each recursion step, the top downtechnique is used for constructing the class‐cquotient Lc(r)of an infinite limit group L(r). Finally, the isomorphism Mc(r)Lc(r)is proved.

Theorem 3.2(An infinite periodically repeating pattern.) Letur=30be an upper bound. An infinite path is generated recursively, since for each2r<ur, the immediate descendantM2r+1(r+1)=M2r(r)#2;1of step size2of the second mainline vertexM2r(r)of the coclass treeT(r)(M2r1(r))is root of a new coclass treeT(r+1)(M2(r+1)1(r+1)). The pruned coclass trees

T(r)(M2r1(r))T(2)(M3(2))

are isomorphicinfinitegraphs, for each2rur. Note that the nuclear rankn(M2r(r))=2.

This is the first main theorem of the present chapter. The proof will be conducted with the aid of an infinite limit group L±, due to M. F. Newman. Certain quotients of L±give precisely the mainline vertices Mc(r)with r2and c2r1as will be shown in Theorem 3.3 and Remark 3.2.

Conjecture 3.1.Theorem 3.2 remains true for any upper bound ur>30.

### 3.4. Mainlines of the pruned descendant tree T(R)

Definition 3.2.The complete theory of the mainlines in T(R)is based on the group

L±:= a,t | (at)3=a3, [[t,a],t]=a±3 |.E3

For each r2, quotients of L±are defined by

L±(r):=L± /  a3r .E4

For each r2, and for each c2r1, quotients of L±(r)are defined by

L±,c(r):={L±(r) /  [t,a]3 if c=2+1 odd, r1,L±(r) /  t3 if c=2 even, r.E5

The following Algorithm 3.1 is based on iterated applications of the p‐group generation algorithm by Newman [7] and O'Brien [8]. It starts with the root R, given by its compact presentation, and constructs an initial section of the unique infinite maintrunk with strictly alternating step sizes 1and 2in the pruned descendant tree T(R). In each step, the required selection of the child with appropriate transfer kernel type (TKT) is achieved with the aid of our own subroutine IsAdmissible(), which is an elaborate version of ([9], section 4.1, p. 76). After reaching an assigned coclass r = hb+2, our algorithm navigates along the mainline of the coclass tree T(r)T(R)and tests each vertex for isomorphism to the corresponding quotient L±,c(r)of class c ≤ 2r − 1+vb.

Algorithm 3.1.(Mainline vertices.)

Input:primep, compact presentationcpof the root, boundshb,vb, signs.

r := 2; // initial coclass

Root := PCGroup(cp);

for i in [1..hb] do // bottom up in double steps along the maintrunk

Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=[1]);

for j in [1..#Des] do

Root := Des[j];

end if;

end for;

r := r + 1; // coclass recursion

Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=[2]);

for j in [1..#Des] do

Root := Des[j];

end if;

end for;

end for;

c := 2*r ‐ 1; // starting class c in dependence on the coclass r

er := pˆr; l := (c ‐ 1) div 2; ec := pˆl;

M<a,t> := Group<a,t|(a*t)ˆp=aˆp,((t,a),t)=aˆ(s*p),aˆer=1,(t,a)ˆec=1>;

QM,pM := pQuotient(M,p,c); // top down construction

if IsIsomorphic(Root,QM) then // identification

printf “Isomorphism for cc=%o, cl=%o.\n”,r,c;

end if;

for i in [1..vb] do // bottom up in single steps along a mainline

c := c + 1; // nilpotency class recursion

if (0 eq c mod 2) then // even nilpotency class

l := c div 2; ec := pˆl;

M<a,t> := Group<a,t|(at)ˆp=aˆp,((t,a),t)=aˆ(sp),aˆer=1,tˆec=1>;

else // odd nilpotency class

l := (c ‐ 1) div 2; ec := pˆl;

M<a,t>:= Group<a,t|(at)ˆp=aˆp,((t,a),t)=aˆ(sp),aˆer=1,(t,a)ˆec=1>;

end if;

QM,pM := pQuotient(M,p,c); // top down construction

Des := Descendants(Root,NilpotencyClass(Root)+1: StepSizes:=[1]);

for j in [1..#Des] do

Root := Des[j];

end if;

end for;

if IsIsomorphic(Root,QM) then // identification

printf “Isomorphism for cc=%o, cl=%o.\n”,r,c;

end if;

end for;

Output:coclassrand classcin each case of an isomorphism.

Remark 3.1.Algorithm 3.1 is designed to be called with input parameters the primep=3andcpthe compact presentation of either the root 243,6with signs=‐1or the root 243,8with sign s=+1. In the current version V2.22‐7 of the computational algebra system MAGMA [10], the bounds are restricted tor=hb+2≤8andc=vb+2r−1≤35, since otherwise the maximal possible internal word length of relators in MAGMA is surpassed. Close to these limits, the required random access memory increases to a considerable value of approximately 8 GB RAM.

Theorem 3.3.(Mainline vertices as quotients of the limit groupL±.) Letur:=8,uc:=35.

1. For each2rur, and for each2r1cuc, the mainline vertexMc(r)of coclassrand nilpotency classcin the treeT(R)is isomorphic toL±,c(r).

2. For each2rur, the projective limit of the mainline(Mc(r))c2r1with vertices of coclassrin the treeT(R)is isomorphic toL±(r).

3. L±is an infinite nonnilpotent profinite limit group.

Proof.(1) The repeated execution of Algorithm 3.1 for successive values fromhb:=0tohb:=6, with input datap:=3,cp:=CompactPresentation(SmallGroup(243,i)), i{6,8}, s{1,+1}, andvb:=32, proves the isomorphisms Mc(r)L±,c(r)for 2rur=8and 2r1cuc=35. The algorithm is initialized by the starting group R=M3(2)=243,iof coclassr:=2. The first loop moves along the maintrunk recursively with strictly alternating step sizes 1and 2until the root M2r1(r)of the coclass tree T(r)withr = 2+hbis reached. The second loop iterates through the mainline vertices Mc(r), c2r1, of the coclass tree T(r)(M2r1(r)), always checking for isomorphism to the appropriate quotient L±,c(r). The subroutineIsAdmissible()tests the transfer kernel type of all descendants and selects the unique capable descendant with type c.18, respectively, c.21. (2) Since periodicity sets in for 2ur1=17cuc=35, the claim is a consequence of Theorem 3.1. (3) The quotient L±(1)is already infinite and nonnilpotent. Adding the relation [t,ta,t]=1suffices to give [t,a,t] central and L±profinite.              □

Conjecture 3.2.Theorem 3.3 remains true for arbitrary upper bounds ur>8, uc>35.

Remark 3.2.When the top down constructions in Algorithm 3.1 are cancelled, the bottom up operations are still able to establish much bigger initial sections of the infinite maintrunk and of the infinite coclass tree with fixed coclass r2. Admitting an increasing amount of CPU time, we can easily reach astronomic values of the coclass, r=32, and the nilpotency class, c=63, that is a logarithmic order of r+c=95, without surpassing any internal limitations of MAGMA, and the required storage capacity remains quite modest, i.e., clearly below 1 GB RAM. This remarkable stability underpins Conjecture 3.2 with additional support from the bottom up point of view.

### 3.5. Covers of metabelian 3‐groups

Only one of the coclass subtrees T(r), r2, of the entire rooted in‐tree T(R)contains metabelian vertices, namely the first subtree T(2). The following theorem shows how transfer kernel types are distributed among metabelian vertices Gof depth dp(G)1on the tree T(2), as partially illustrated by the Figures 1 and 2.

Theorem 3.4.(Metabelian vertices of the coclass treeT(2)R.)

For each finite3‐groupG, we denote byc:=cl(G)the nilpotency class, byr:=cc(G)the coclass, and byϰthe transfer kernel type ofG. More explicitly, such a group is also denoted byG=Gc(r). The following statements describe the structure of the metabelian skeleton of the coclass treeT(2)Rwith rootR:=243,6, respectively,R:=243,8, down to depth1.

1. For eachc3, the mainline vertexMc(2)of the coclass tree possesses typec.18,ϰ=(0122), respectively,c.21,ϰ=(0231).

2. For eachc4, there exists a unique childGc,1(2)ofMc1(2)with typeE.6,ϰ=(1122), respectively,E.8,ϰ=(1231).

3. For evenc4, there exists a unique childGc,2(2)ofMc1(2)with typeE.14,ϰ=(3122), respectively,E.9,ϰ=(2231). Thus,N1(Mc1(2))=3andC1(Mc1(2))=1, in the pruned tree.

4. For oddc5, there exist two childrenGc,2(2),Gc,3(2)ofMc1(2)with typeE.14,ϰ=(3122)(4122), respectively,E.9,ϰ=(2231)(3231). Thus,N1(Mc1(2))=4andC1(Mc1(2))=1.

5. For evenc4, there exists a unique childGc,4(2)ofMc1(2)with typeH.4,ϰ=(2122), respectively,G.16,ϰ=(4231). It is removed from the pruned tree.

6. For oddc5, there exist two childrenGc,4(2),Gc,5(2)ofMc1(2)with typeH.4,ϰ=(2122), respectively,G.16,ϰ=(4231). They are removed from the pruned tree.

Proof.See Nebelung ([11], Lemma 5.2.6, p. 183, Figures, p. 189 f., Satz 6.14, p. 208).     □

Definition 3.3.For e{0,1}, we define the cover limit, due to M. F. Newman, to be the group

C(e):= a,t,u,y,z  |  ta=u, uatuy=[u,t]e, a3[t,a,t]=z, [u,t,t]=[u,t,u]=1,y3=1, [a,y]=[t,y]=[u,y]=[z,y]=1, z3=1, [t,z]=[u,z]=1 ,E6

which was introduced in [12]. For each k{1,0,1}and for each integer c4, let

Qc(e,k):=C(e) /  ywckvc, zwc E7

be the class‐c quotient with parameterkof C(e), where wc:=[t,a,,a](c1)timesand vc:=[wc2,[t,a]].

In each step, i1, of the second Algorithm 3.2, the top downtechnique constructs a certain class‐cquotient Qc, c=i+3, of a fixed infinite pro‐3 group C, the cover limit, and the bottom uptechnique constructs all metabelian children of a certain vertex Mi1on the mainline of the first coclass tree T(2)(R), and selects, firstly, the next vertex Miof depth dp(Mi)=0on the mainline of T(2)(R)for continuing the recursion, secondly, a vertex Giof depth dp(Gi)=1with assigned transfer kernel type ϰ(Gi). Each recursion step is completed by proving that Giis isomorphic to the second derived quotient Qc/Qc, that is, Qccov(Gi)belongs to the coverof Giin the sense of ([13], section 1.3, Dfn. 1.1, p. 75). More precisely, we have Mi=Mi+3(2)and Gi=Gi+3,j(2)with some j.

Algorithm 3.2.(Shafarevich cover.)

Input:primep, compact presentationcpof the root, boundvb, parameterseandk.

C<a,t,u,y,z> := Group< a,t,u,y,z |

yˆp, (a,y), (t,y), (u,y), (y,z), (t,z), (u,z), zˆp,

(u,t,t), (u,t,u), tˆa = u, uˆatuy(u,t)ˆ‐e, aˆp(t,a,t) = z >;

Root := PCGroup(cp);

Leaf := Root;

for i in [1..vb] do // bottom up along the mainline of coclass 2

c := i + 3; // nilpotency class

w := [t];

for j in [1..c] do // construction of iterated commutator

s := (w[j],a);

Append(w,s);

end for;

w1 := w[c‐2]ˆ‐1(a,t)w[c‐2](t,a);

H := quo<C | yw[c]ˆkw1, zw[c]>;

Q,pQ := pQuotient(H,p,c); // top down construction of Shafarevich cover

Des := Descendants(Root,NilpotencyClass(Root)+1);

m := 0;

for cnt in [1..#Des] do

Root := Des[cnt]; // next mainline vertex

m := m + 1;

if (1 eq m) then

Leaf := Des[cnt]; // first leaf with assigned TKT

end if;

end if;

end for;

DQ := DerivedSubgroup(Q);

D2Q := DerivedSubgroup(DQ);

Q2Q := Q/D2Q; // metabelianization

if IsIsomorphic(Leaf,Q2Q) then // identification

printf “Dsc.cl.%o isomorphic to 2nd drv.qtn.of Cov.cl.%o.\n”,c,c;

end if;

end for;

Output:nilpotency class c in each case of an isomorphism.

The next theorem is the second main result of this chapter, establishing the finiteness and structure of the coverfor each metabelian 3‐group with transfer kernel of type E.

Theorem 3.5.(Explicit covers of metabelian 3‐groups.) Letu:=8be an upper bound andGc,j(2)inT(2)(M3(2))be the metabelian 3‐group of nilpotency classc4with transfer kernel type

ϰ={(1122), E.6, resp. (1231), E.8if j=1,(3122), E.14, resp. (2231), E.9if j=2 or (j=3 and c odd).
1. ThecoverofGc,j(2)is given by

cov(Gc,j(2))={{Gc,j(2);Gc,j(3),,Gc,j(+1),Gc,j(+2),Tc+1,j(+2)}if c=2+4, 1j2,{Gc,j(2);Gc,j(3),,Gc,j(+1),Gc,j(+2),Sc,j(+3)}if c=2+5, 1j3.E8

where0u. In particular, the cover is a finite set with+2elements (+1of them nontrivial), which are nonσ‐groups for evenc4, and σ‐groups for oddc5.

2. TheShafarevich coverofGc,j(2)with respect to imaginary quadratic fieldsFis given by

cov(Gc,j(2),F)={øif c=2+4, 0u, 1j2,{Sc,j(+3)}if c=2+5, 0u, 1j3.E9

In particular, the Shafarevich cover contains a unique Schur σ‐group, ifc5is odd.

3. The class‐c quotient with parameterkof the cover limitC(e)is isomorphic to a Schur σ‐groupQc(e,k)Sc,j(+3), forc=2+5or to a nonσ‐groupQc(e,k)Gc,j(+2), forc=2+4. The precise correspondence between the parameterskandjis given in the following way.

Types E.6,E.8: Qc(e,0){Sc,1(+3)for odd class c=2+5, 0u,Gc,1(+2)for even class c=2+4, 0u,type E.9: Qc(+1,1){Sc,2(+3)for odd class c=2+5, 0u,Gc,2(+2)for even class c=2+4, 0u,type E.9: Qc(+1,+1){Sc,3(+3)for odd class c=2+5, 0u,Gc,2(+2)for even class c=2+4, 0u.E10

In particular,Qc(+1,1)Qc(+1,+1)for even classc=2+4,0u.

The variante=0, respectively,e=1, is associated to the rootR=243,6, respectively,R=243,8.

4. A parameterized family offork topologiesfor second3‐class groupsGal(F3(2)/F)of imaginary quadratic fieldsFis given uniformly for the states(ground state for=0, excited state for1u) of transfer kernels with typeEby the symmetric topology symbol

P=E1Leaf{c1}2 Mainline  c  Fork{2c1c} Maintrunk2ELeafE11

with scaffold typecand the following invariants:

distanced=4+2(Definition 2.3), weighted distancew=5+3(Definition 2.4),

class incrementΔcl=(2+5)(2+5)=0, coclass incrementΔcc=(+3)2=+1,

logarithmic order incrementΔlo=(3+8)(2+7)=+1([13], Dfn. 5.1, p. 89).

Proof.We compare the uniform generator rank d1=2of all involved groups Gc,j(r), c4, r2, 1j3, with their relation rank d2. Since d2=μand the p‐multiplicator rank is μ=2for Sc,j(r)with odd c=2+55and r=+33, but μ=3otherwise, only the groups Sc,j(r)are Schur σ‐groups with balanced presentation d2=2=d1, and are therefore admissible as 3‐tower groups of imaginary quadratic fields F, according to our corrected version ([6], section 5, Thm. 5.1, pp. 28–29) of the Shafarevich Theorem ([14], Thm. 6, (18′)). Finally, we remark that the nuclear rank is ν=1for Gc,j(r)with even c=2+4, r=+2, and child Tc+1,j(r), but ν=0otherwise.

The execution of Algorithm 3.2 with input datap:=3, vb:=25, eitheri:=6, e:=0, ori:=8, e:=1, andcp:=CompactPresentation(SmallGroup(243,i)), proves the isomorphisms Qc(e,k)Sc,j(+3), c=2+5, respectively, Qc(e,k)Gc,j(+2), c=2+4, for 4c20, that is, 0u=8. The algorithm is initialized by the starting group R=M3(2)of coclass 2. The loop navigates through the mainline vertices Mc(2), c3, of the coclass tree T(2)(M3(2)). The subroutineIsAdmissible()tests the transfer kernel type of all descendants and selects either the unique capable descendant with type c.18, respectively, c.21, for the flag 0 or the unique descendant with type E.6, respectively, E.8, for the flag 1, or the first or second descendant with type E.9, for the flag 2. The selected nonmainline vertex is always checked for isomorphism to the metabelianization of the appropriate quotient Qc(e,k). See also ([2], section 21.2, pp. 189–193), ([15], pp. 751–756), the proof of Theorem 4.1, and Figures 57.

Here again, a pure bottom up approach without top down constructions, instead of using Algorithm 3.2, is able to reach coclass r=32, nilpotency class c=63, and logarithmic order r+c=95, without surpassing internal limits of MAGMA, and strongly supports Conjecture 3.3.

Conjecture 3.3.Theorem 3.5 remains true for any upper bound u>8.

Figure 3 shows exactly the same situation as Figure 1, supplemented by blue arrows indicating the projections of the quotients Qc(e,k)onto their metabelianizations, that is, Sc,j(+3)Gc,j(2), for odd class c=2+5, in the right diagram with green branches, and Gc,j(+2)Gc,j(2), for even class c=2+4, in the left diagram with red branches. For c=4, a degeneration occurs, since Q4(e,k)is metabelian already, indicated by surrounding blue circles.

Strictly speaking, the caption of Figure 3, in its full generality, is valid for e=1, M3(2)=243,8only. For e=0, M3(2)=243,6, all blue arrows have the same meaning as before but the interpretation of the covers as quotients Qc(e,k)is slightly restricted. Whereas we have the following supplement to Eq. (10):

type E.14: Qc(0,1){Sc,3(+3)for odd class c=2+5, 0u,Gc,2(+2)for even class c=2+4, 0u,E12

the quotients Qc(0,+1)lead into a completely different realm, namely the complicated brushwood of the complex transfer kernel type H.4.

Figure 4 shows three pruned descendant trees T()with roots =243,4, =6561,614, and =6561,613#1;1#2;1, all of whose vertices are of type H.4exclusively. We restrict the trees to σ‐groups indicated by green color. The top vertex 27,3is intentionally drawn twice to avoid an overlap of the dense trees and to admit a uniform representation of periodic bifurcations.

The tree with root 243,4is not concerned by the quotients Qc(0,+1). It is sporadic and consists of periodically repeating finite saplings of depth 2and increasing coclass 2,3,. Connected by the maintrunk with vertices of type c.18(red color) in the descendant tree T(243,6), the trees with roots 6561,614and 6561,613#1;1#2;1form the beginning of an infinite sequence of similar trees, which are, however, not isomorphic as graphs, since the depth of the constituting saplings increases in steps of 2. The projections of the quotients Qc(0,+1)with odd class c{5,7}onto their metabelianizations are indicated by blue arrows.

### 3.6. Topologies in descendant trees

Tree topologies describe the mutual location of distinct higher p‐class groups Gp(m)Fand Gp(n)F, with n>m1, of an algebraic number field F. The case (m,n)=(3,4)will be crucial for finding the first examples of four‐stage towersof p‐class fields with length pF:=dl(Gp()F)=4, which are unknown up to now, for any prime p2. Fork topologies with (m,n)=(2,3)have proved to be essential for discovering p‐class towers with length pF=3, for odd primes p3. In ([13], Prp. 5.3, p. 89), we have pointed out that the qualitativetopology problem for (m,n)=(1,2)is trivial, since the fork of Gp(1)Fand Gp(2)Fis simply the abelian root Gp(1)FClpFof the entire relevant descendant tree. However, the quantitativestructure of the root path between Gp(2)Fand Gp(1)Fis not at all trivial and can be given in a general theorem for ClpF(p,p)and p{2,3}only. In the following Theorem 3.6, we establish a purely group theoretic version of this result by replacing Gp(2)Fwith an arbitrary metabelian3‐group Mhaving abelianization M/Mof type (3,3). Any attempt to determine the group G:=Gal(Fp()/F)of the p‐class tower Fp()of an algebraic number field Fbegins with a search for the metabelianization M:=G/G, i.e., the second derived quotient, of the p‐tower group G. Mis also called the second p‐class groupGal(Fp(2)/F)of F, and Fp(2)can be viewed as a metabelian approximation of the p‐class tower Fp(). In the case of the smallest odd prime p=3and a number field Fwith 3‐class group Cl3Fof type (3,3), the structure of the root path from Mto the root 9,2is known explicitly. For its description, it suffices to use the set of possible transfer kernel types

X{A,D,E,F,G,H,a,b,c,d}

of the ancestors πjM, 0j, and the symbol sfor a weighted edge of step size s1with formal exponents denoting iteration. A capable vertex is indicated by an asterisk X.

Theorem 3.6. (Periodic root paths.)

There exist basically three kinds of root pathsP:=(πjM)0jof metabelian 3‐groupsMwith abelianizationM/Mof type(3,3), which are located on coclass trees. Letcdenote the nilpotency classcl(M)andrthe coclasscc(M)ofM.

1. Ifr=1andc1, thenP=X{1a}c1, whereX{A,a,a}.

2. Ifr=2andc3, then eitherP=X{1b}c32a1a, whereX{d,b,b}, orP=X{1c}c32a1a, whereX{E,G,H,c}. An additional variant arises forr=2, c5, withP=X1X{1c}c42a1a, whereX{G,H}.

3. Ifr3andcr+1, then eitherP=X{1b}c(r+1){2b}r22a1a, whereX{d,b,b}, orP=X{1d}c(r+1){2b}r22a1a, whereX{F,G,H,d}. An additional variant arises forr3, cr+3, withP=X1X{1d}c(r+2){2b}r22a1a, whereX{G,H}.

In particular, the maximal possible step size iss=2, and ther1edges with step sizes=2arise successively without gaps at the end of the path, except the trailing edge of step sizes=1.

Proof.Xalways denotes the type of the starting vertex M. The remaining vertices of the root path form the scaffold, which connects the starting vertex with the ending vertex (the root R=9,2). The unique coclass tree T(1)9,2with r=1has a mainline of type a. Two of the coclass trees T(2)243,nwith r=2, those with n{6,8}, have mainlines of type cand an additional scaffold of type a. For n=3, the mainline is of type b. The coclass trees T(r)with r3behave uniformly with mainlines of type bor dand scaffold types b, a. For details, see Nebelung ([11], Satz 3.3.7, p. 70, Lemma 5.2.6, p. 183, Satz 6.9, p. 202, Satz 6.14, p. 208).

Remark 3.3.The final statement of Theorem 3.6 is a graph theoretic reformulation of the quotient structure of the lower central series (γjM)j1of a metabelian 3‐group M, observing that the root R=9,2corresponds to the bicyclic quotient γ1/γ2(3,3)and the conspicuous trailing edge 1acorresponds to the cyclic bottleneck γ2/γ3(3). The structure is drawn ostensibly in eqn. (2.12) of ([16], section 2.2), using the CF‐invariant e=r+1instead of the coclass r.

Theorem 3.6 concerns periodic vertices on coclass trees. Sporadic vertices outside of coclass trees must be treated separately in Corollary 3.1.

As before, letMbe a metabelian 3‐group with abelianizationM/M(3,3), nilpotency classc:=cl(M), and coclassr:=cc(M). Assume thatMis located outside of coclass trees.

1. Ifr=2andc=3, thenP=X2a1a, whereX{D,G,H}.

2. Ifr=2andc=4, thenP=X1X2a1a, whereX{G,H}.

3. Ifr3andc=r+1, thenP=X{2b}r22a1a, whereX{F,G,H}.

4. Ifr3andc=r+2, thenP=X1X{2b}r22a1a, whereX{G,H}.

Proof.As in the proof of Theorem 3.6, see the dissertation of Nebelung [11].        □

### 3.7. Computing Artin patterns of p‐groups

In both Algorithms 3.1 and 3.2, we made use of a subroutineIsAdmissible()which filters p‐groups Gwith abelianization G/G(p,p)having a prescribed transfer kernel type (TKT). Since an algorithm of this kind is not implemented in MAGMA, we briefly communicate a succinct form of the code for this subroutine.

Algorithm 3.3.(Transfer kernel type.)

Input:a prime number pand a finite p‐group G.

Code:

if ([p,p] eq AbelianQuotientInvariants(G)) then

x := G.1; y := G.2; // main generators

A := []; B := []; // generators and transversal

Append(A,y);

Append(B,x);

for e in [0..p‐1] do

Append(A,xyˆe);

Append(B,y);

end for;

DG := DerivedSubgroup(G);

nTotal := 0; nFixed := 0;

TKT := [];

for i in [1..p+1] do

M := sub<G|A[i],DG>;

DM := DerivedSubgroup(M);

AQM,pr := M/DM;

ImA := (A[i]B[i]ˆ‐1)ˆpB[i]ˆp; // inner transfer

ImB := B[i]ˆp; // outer transfer

T := hom<G‐>AQM|<A[i],(ImA)@pr>,<B[i],(ImB)@pr>>;

KT := sub<G|DG,Kernel(T)>;

if KT eq G then // total kernel

Append(TKT,0);

nTotal := nTotal+1;

else

for j in [1..p+1] do

if A[j] in KT then

Append(TKT,j);

if (i eq j) then // fixed point

nFixed := nFixed+1;

end if;

end if;

end for;

end if;

end for;

image := [];

for i in [1..p+1] do

if not (TKT[i] in image) then

Append(image,TKT[i]);

end if;

end for;

occupation := #image;

repetitions := 0; // maximal occupation number

intersection := 0; // meet of repetitions and fixed points

doublet := 0;

for digit in [1..p+1] do

counter := 0;

for j in [1..#TKT] do

if (digit eq TKT[j]) then

counter := counter + 1;

end if;

end for;

if (counter ge 2) then

doublet := digit;

end if;

if (counter gt repetitions) then

repetitions := counter;

end if;

end for;

if (doublet ge 1) then

if (doublet eq TKT[doublet]) then

intersection := 1;

end if;

end if;

end if;

Output:transfer kernel typeTKT, numbernTotalof total kernels, numbernFixedof fixed points, and further invariantsoccupation,repetitions,intersectiondescribing the orbit of the TKT.

The output of Algorithm 3.3 is used for the subroutineIsAdmissible(G,p,t)in dependence on the parameter flag t. When the root R=243,8is selected for the tree T(R)the return value is determined in the following manner

if (0 eq t) then

return ((1 eq nTotal) and (2 eq nFixed)); // type c.21

elif (1 eq t) then

return ((0 eq nTotal) and (3 eq nFixed)); // type E.8

elif (2 eq t) then

return ((0 eq nTotal) and (2 eq nFixed) and (3 eq occupation)); // type E.9

end if;

For the root R=243,6, we have

if (0 eq t) then

return ((1 eq nTotal) and (0 eq nFixed)); // type c.18

elif (1 eq t) then

return ((0 eq nTotal) and (1 eq nFixed)); // type E.6

elif (2 eq t) then

return ((0 eq nTotal) and (0 eq nFixed) and (3 eq occupation)); // type E.14

end if;

### 3.8. Benefits and drawbacks of bottom up and top down techniques

In this chapter, we have presented several convenient ways of expressing information about infinite sequencesof finite p‐groups. Each of them has its benefits and drawbacks.

The bottom up strategyof constructing finite p‐groups as successive extensions of a (metabelian or even abelian) starting group R, called the root, by recursive applications of the p‐group algorithm by Newman [7] and O'Brien [8] has the benefit of visualizing the graph theoretic root pathin the descendant tree T(R). Its implementation in MAGMA is incredibly stable and robust without surpassing any internal limits up to logarithmic orders of 95and even more. Only the consumption of CPU time becomes considerable in such extreme regions.

The top down strategyof expressing finite p‐groups as quotients of an infinite pro‐pgroup with given pro‐ppresentation has the benefit of including nonmetabelian groups with arbitrary coclass r3, periodic mainline vertices in Algorithm 3.1, and sporadic Schur σ‐leaves in Algorithm 3.2. The drawback is that the evaluation of the pro‐ppresentation in MAGMA exceeds the maximal permitted word length for nilpotency class c36.

Up to this point, we have not yet touched upon parameterized polycyclic power‐commutator presentations[17]. For the root R=243,6, the metabelian vertices Gof the coclass tree T(2)(R)with class c=cl(G)5, down to depth dp(G)1, can be presented in the form

Gc(α,β)=x,y,s2,t3,s3,,sc |s2=[y,x], t3=[s2,y], si=[si1,x] for 3ic,x3=scα, y3s32s41=scβ, si3=si+22si+3 for 2ic3, sc23=sc2 ,E13

where the parameters αand βdepend on the transfer kernel type ϰ(G),

(α,β)={(0,0)for ϰ(G)(0122), c.18,(1,0)for ϰ(G)(1122), E.6,(0,1) or (0,2)for ϰ(G)(2122), H.4,(1,1) or (1,2)for ϰ(G)(3122)(4122), E.14.E14

This presentation has the benefit of including six periodic sequences with distinct transfer kernel types, and the drawback of being restricted to the fixed coclass 2.

## 4. The first 3‐class towers of length 3

In our long desired disproof of the claim by Scholz and Taussky ([18], p. 41) concerning the 3‐class tower of the imaginary quadratic field F=Q(9748), we presented the first p‐class towers with exactly three stages, for an odd prime p, in cooperation with Bush ([19], Cor. 4.1.1, p. 775). The underlying fields Fwere of type E.9in its ground state, which admits two possibilities for the second 3‐class group, M2187,302or 2187,306. Now we want to illustrate the way which led to the fork topologiesin Theorem 3.5 by using the more convenient type E.8, where the group Mis unique for every state, in particular, M2187,304for the ground state.

Remark 4.1.Concerning the notation, we are going to use logarithmic type invariantsof abelian 3‐groups, for instance (21)=^(9,3), (32)=^(27,9), (43)=^(81,27), and (54)=^(243,81).

Let F=Q(d)be an imaginaryquadratic number field with 3‐class group Cl3F(3,3), and let E1,,E4be the unramified cyclic cubic extensions of F.

Theorem 4.1.(First towers of typeE.8.) Let the capitulation of 3‐classes ofFinE1,,E4be of typeϰ1F(1,2,3,1), which is called typeE.8. Assume further that the 3‐class groups ofE1,,E4possess the abelian type invariantsτ1F[T1,21,21,21], whereT1{32,43,54}.

Then, the length of the3‐class tower ofFis precisely3F=3.

Proof.We employ the p‐group generation algorithm [7, 8] for searching the Artin pattern AP(F)=(τ1F,ϰ1F)among the descendants of the root R:=C3×C3=9,2in the tree T(R). After two steps, 9,227,3243,8, we find the next root U5:=243,8of the unique relevant coclass tree T(2)(U5), using the assigned simple TKT E.8, ϰ3=(1231), and its associated scaffold TKT c.21, ϰ0=(0231). Finally, the first component T1=τ1(1){32,43,54}of the TTT provides the break‐off condition, according to ([13], Thm. 1.21, p. 79), respectively, Theorem M in ([20], p. 14), and we get M2187,304=729,54#1;4for the ground state T1=(32), M6561,2050#1;2for the first excited state T1=(43), and M6561,2050(#1;1)2#1;2for the second excited state T1=(54), where 2187,303=729,54#1;3and 6561,2050=2187,303#1;1. The situation is visualized by Figure 2, where the three metabelianizations MG/Gof the 3‐tower group G, for the ground state and two excited states, are emphasized with red color. Figure 2, showing the second 3‐class groups M, was essentially known to Ascione in 1979 [21, 22], and to Nebelung in 1989 [11]. Compare the historical remarks ([2], section 3, p. 163).

In the next three Figures 57, which were unknown until 2012, we present the decisive break‐through establishing the first rigorous proof for three‐stage towers of 3‐class fields. The key ingredient is the discovery of periodic bifurcations ([2], section 3, p. 163) in the complete descendant tree T(U5)which is of considerably higher complexity than the coclass tree T(2)(U5).

For the ground state T1=(32), the first bifurcation yields the cover

cov(M)={M,6561,622}

of M2187,304. The relation rank d2M=3eliminates Mas a candidate for the 3‐tower group G, according to the Corollary ([20], p. 7) of the Shafarevich Theorem ([13], Thm. 1.3, pp. 75–76), and we end up getting G6561,622=729,54#2;4with a siblings topology

E1 c 2E

which describes the relative location of Mand G.

For the first excited state T1=(43), the second bifurcation yields the cover

cov(M)={M,6561,621#1;1#1;2,6561,621#1;1#2;2}

of M6561,2050#1;2, where 6561,621=729,54#2;3. The relation rank d2=3eliminates Mand 6561,621#1;1#1;2as candidates for the 3‐tower group G, according to Shafarevich, and we get the unique G6561,621#1;1#2;2with fork topology

E1 {c1}2c {2c1c}2E.

Similarly, the second excited state T1=(54)yields a more complex advanced fork topology

E1 {c1}4c {2c1c}22E.

Figure 5 impressively shows that entering the unnoticed secret door, which is provided by the bifurcation at the vertex 729,54, immediately leads to the long desired 3‐tower group G6561,622=729,54#2;4of the imaginary quadratic field F=Q(34867). The siblings topology is emphasized with red color, and the projection GMG/Gis drawn in blue color.

In Figure 6, we see that the path to the 3‐tower group G729,54#2;3#1;1#2;2of the imaginary quadratic field F=Q(370740)contains two bifurcations at 729,54and 729,54#2;3#1;1. As before, the fork topology is emphasized with red color, and the projection GMG/Gis drawn in blue color. Two projection arrows of type E.9are black.

Figure 7 shows the path to the 3‐tower group G729,54#2;3#1;1#2;1#1;1#2;2of the imaginary quadratic field F=Q(4087295). It requires three bifurcations at 729,54, 729,54#2;3#1;1, and 729,54#2;3#1;1#2;1#1;1. Again, the fork topology is emphasized with red color, and the projection GMG/Gis drawn in blue color.

## 5. Future developments

Fork topologies with significantly higher complexity and step sizes up to 3and even 4will be investigated in cooperation with M. F. Newman [23] for finite 3‐groups with TKT F.

## Acknowledgments

The author gratefully acknowledges support from the Austrian Science Fund (FWF): P 26008‐N25. Indebtedness is expressed to M. F. Newman from the Australian National University, Canberra, for valuable help in providing evidence of Theorems 3.2, 3.3 and 3.5.

## References

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Written By

Daniel C. Mayer

Submitted: October 18th, 2016 Reviewed: March 22nd, 2017 Published: December 20th, 2017