Open access peer-reviewed chapter

Modeling Rooted in‐Trees by Finite p‐Groups

Written By

Daniel C. Mayer

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

DOI: 10.5772/intechopen.68703

From the Edited Volume

Graph Theory - Advanced Algorithms and Applications

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 multifurcation in 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:GπG 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)). Since the structure of our rooted in‐trees is rather complex, we use pattern recognition methods 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 analysis for 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 periodicity of pruned coclass subtrees eventually admits the reduction of the complete infinite graph T(R) to finite patterns. Evidence of these newly discovered periodic bifurcations is provided by a mixture of bottom up techniques, using successive extensions by means of p‐covering groups in the pgroup generation algorithm and top down techniques using infinite limits and their finite quotients in Sections 3.4 and 3.5. As a coronation of this chapter, we show in Sections 3.6 and 4 that fork topologies provide 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 towers Fp() of algebraic number fields F.

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2. Underlying abstract graph theory

Let G=(V,E) be a graph with set of vertices V and set of edges E. We expressly admit infinite sets V and 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×V than only subsets {v1,v2}V with two elements. Such a directed edge e=(v1,v2) is also denoted by an arrow v1v2 with starting vertex v1 and ending vertex v2. Thus, we have EV×V. Now, infinitude comes in.

Definition 2.1. (Finite and infinite paths.)

A finite path of length 0 in G is a finite sequence (vi)0i of vertices viV such that (vi,vi+1)E for 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 path of length =0.

An infinite path in G is an infinite sequence (vi)i0 of vertices viV such that (vi+1,vi)E for all i0. In this case, v0 is 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‐trees G=T(R), that is, connected digraphs without cycles such that the root vertex R has out‐degree 0, whereas any other vertex vV\{R} has out‐degree 1. A vertex with in‐degree at least 1 is called capable, whereas a vertex with in‐degree 0 is 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)E is the unique edge with starting vertex v, is called the parent operator of T(R). For each vertex vV, there exists a unique finite root path from v to the root R,

v=π0vπ1vπ2vπ1vπv=R,

expressed by iterated applications of the parent operator and with some length 0. Each vertex in the root path of v is called an ancestor of v.

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

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

We can define a partial order on the vertices u,aV of the tree T(R) by putting ua if uT(a), that is, if u is descendant of a and a is ancestor of u. The root R is the minimum.

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

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

2.3. Mainlines and multifurcation

We shall also need weight functions with 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 V0 is precisely the set of leaves of the tree T(R). Thus, there arise weighted measures.

Definition 2.4. (Path weight and weighted distance.)

By the path weight of a finite path (vi)0i with length 0 in T(R) such that (vi,vi+1)Esi for 0i1, we understand the sum i=01si. The sum wu+wv of the path weights from two vertices u,vV to their fork is called the weighted distance w(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)i0 in T(R) with edges of weight 1, that is, such that (vi+1,vi)E1 for all i0, is called a mainline in T(R).

The minimal tree T1(a)=(V1(a),E1(a)) of a vertex aV is the subtree of the descendant tree T(a)=(V(a),E(a)) consisting of vertices v, whose root path in T(a) possesses edges e of weight 1 only, 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)i0 be a mainline in T(R). For i0, the difference set B(vi):=T1(vi)\T1(vi+1) of minimal trees is called the branch with root vi of 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 n2 be a positive integer. A vertex aVn has an n‐fold multifurcation if its in‐degree is an n‐fold sum N1+N2++Nn due to Ns1 incoming edges of weight s, for each 1sn. That is, we define counters Ns of all incoming edges of weight s, and additionally, we have counters Cs of all incoming edges of weight s with 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 a by writing a#s;i for the ith immediate descendant with edge of weight s, where 1sn and 1iNs.

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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 vV are represented by isomorphism classes of finite p‐groups G, for a fixed prime number p. Directed edges eE are represented by epimorphisms π:GπG of finite p‐groups with characteristic kernels ker(π)=γcG, where c:=cl(G) denotes the nilpotency class of G and (γiG)i1 is the lower central series of G.

We emphasize that the symbol π is 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, πG is the parent of G.

The weight of a vertex G is realized by its nuclear rank n(G) ([2], section 14, eqn. (28), p. 178) and the weight of a directed edge π:GπG is realized by its step size s(π)=logp(#γcG) ([2], section 17, eqn. (33), p. 179). These invariants are essential for understanding the phenomenon of multifurcation in Definition 2.7. In particular, we can hide multifurcation by restricting all edges π to 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 G of 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 order lo(G):=logp(ord(G)) of G. Generally, the logarithmic order of an immediate descendant G with parent πG increases 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 G is 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:=243,6 or 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=3 for the theory of rooted in‐trees as developed in Section 2, the following finiteness and periodicity statement becomes provable.

The virtual periodicity of 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 π with 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)i0 in T(R), there exists a periodic root vϱ with ϱ0 and a period length λ1 such that the branches

B(vi+λ)B(vi)

are isomorphic finite graphs, for all iϱ. Up to a finite preperiodic component, the minimal tree T1(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=3 and 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.

Figure 1.

Graph dissection into pruned branches connected by the mainline skeleton.

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 σgroup G admits an automorphism σAut(G) acting as inversion σ(x)=x1 on 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 1 and 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:=R of the mainline (vi)i0, and by using the variable class c3 and the fixed coclass r=2 as parameters describing all mainline vertices Mc(r):=vc3. The periodic root is M5(2)=v2 with ϱ=2 and 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=2 and 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) ϰ as its horizontal axis and the first component τ(1) of the transfer target type (TTT) τ as 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 simple types 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 complex type G.16, ϰ=(4231), which are capable at depth 1 and give rise to a complicated brushwood of various descendants (right of the mainline).

Figure 2.

Coclass tree T(2)(243,8) with simple, scaffold, and complex types.

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 d of (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.8 are 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 bifurcations in pruned descendant trees which contain a unique infinite path of edges π with strictly alternating step sizes s(π)=1 and s(π)=2, the so‐called maintrunk. It is very important that the trees are pruned in 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 up technique. In each recursion step, the top down technique is used for constructing the class‐c quotient 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.) Let ur=30 be an upper bound. An infinite path is generated recursively, since for each 2r<ur, the immediate descendant M2r+1(r+1)=M2r(r)#2;1 of step size 2 of the second mainline vertex M2r(r) of the coclass tree T(r)(M2r1(r)) is root of a new coclass tree T(r+1)(M2(r+1)1(r+1)). The pruned coclass trees

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

are isomorphic infinite graphs, for each 2rur. Note that the nuclear rank n(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 r2 and c2r1 as 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 1 and 2 in 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: prime p, compact presentation cp of the root, bounds hb,vb, sign s.

Code: uses the subroutine IsAdmissible().

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

    if IsAdmissible(Des[j],p,0) then

      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

    if IsAdmissible(Des[j],p,0) then

      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

    if IsAdmissible(Des[j],p,0) then

      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: coclass r and class c in each case of an isomorphism.

Remark 3.1. Algorithm 3.1 is designed to be called with input parameters the prime p=3 and cp the compact presentation of either the root 243,6 with sign s=‐1 or the root 243,8 with sign s=+1. In the current version V2.22‐7 of the computational algebra system MAGMA [10], the bounds are restricted to r=hb+2≤8 and c=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 group L±.) Let ur:=8, uc:=35.

  1. For each 2rur, and for each 2r1cuc, the mainline vertex Mc(r) of coclass r and nilpotency class c in the tree T(R) is isomorphic to L±,c(r).

  2. For each 2rur, the projective limit of the mainline (Mc(r))c2r1 with vertices of coclass r in the tree T(R) is isomorphic to L±(r).

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

Proof. (1) The repeated execution of Algorithm 3.1 for successive values from hb:=0 to hb:=6, with input data p:=3, cp:=CompactPresentation(SmallGroup(243,i)), i{6,8}, s{1,+1}, and vb:=32, proves the isomorphisms Mc(r)L±,c(r) for 2rur=8 and 2r1cuc=35. The algorithm is initialized by the starting group R=M3(2)=243,i of coclass r:=2. The first loop moves along the maintrunk recursively with strictly alternating step sizes 1 and 2 until the root M2r1(r) of the coclass tree T(r) with r = 2+hb is 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 subroutine IsAdmissible() 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]=1 suffices 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 G of depth dp(G)1 on the tree T(2), as partially illustrated by the Figures 1 and 2.

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

For each finite 3‐group G, we denote by c:=cl(G) the nilpotency class, by r:=cc(G) the coclass, and by ϰ the transfer kernel type of G. More explicitly, such a group is also denoted by G=Gc(r). The following statements describe the structure of the metabelian skeleton of the coclass tree T(2)R with root R:=243,6, respectively, R:=243,8, down to depth 1.

  1. For each c3, the mainline vertex Mc(2) of the coclass tree possesses type c.18, ϰ=(0122), respectively, c.21, ϰ=(0231).

  2. For each c4, there exists a unique child Gc,1(2) of Mc1(2) with type E.6, ϰ=(1122), respectively, E.8, ϰ=(1231).

  3. For even c4, there exists a unique child Gc,2(2) of Mc1(2) with type E.14, ϰ=(3122), respectively, E.9, ϰ=(2231). Thus, N1(Mc1(2))=3 and C1(Mc1(2))=1, in the pruned tree.

  4. For odd c5, there exist two children Gc,2(2), Gc,3(2) of Mc1(2) with type E.14, ϰ=(3122)(4122), respectively, E.9, ϰ=(2231)(3231). Thus, N1(Mc1(2))=4 and C1(Mc1(2))=1.

  5. For even c4, there exists a unique child Gc,4(2) of Mc1(2) with type H.4, ϰ=(2122), respectively, G.16, ϰ=(4231). It is removed from the pruned tree.

  6. For odd c5, there exist two children Gc,4(2), Gc,5(2) of Mc1(2) with type H.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 parameter k of C(e), where wc:=[t,a,,a](c1)times and vc:=[wc2,[t,a]].

In each step, i1, of the second Algorithm 3.2, the top down technique constructs a certain class‐c quotient Qc, c=i+3, of a fixed infinite pro‐3 group C, the cover limit, and the bottom up technique constructs all metabelian children of a certain vertex Mi1 on the mainline of the first coclass tree T(2)(R), and selects, firstly, the next vertex Mi of depth dp(Mi)=0 on the mainline of T(2)(R) for continuing the recursion, secondly, a vertex Gi of depth dp(Gi)=1 with assigned transfer kernel type ϰ(Gi). Each recursion step is completed by proving that Gi is isomorphic to the second derived quotient Qc/Qc, that is, Qccov(Gi) belongs to the cover of Gi in 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: prime p, compact presentation cp of the root, bound vb, parameters e and k.

Code: uses the subroutine IsAdmissible().

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

     if IsAdmissible(Des[cnt],p,0) then

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

     elif IsAdmissible(Des[cnt],p,2) then

       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 cover for each metabelian 3‐group with transfer kernel of type E.

Theorem 3.5. (Explicit covers of metabelian 3‐groups.) Let u:=8 be an upper bound and Gc,j(2) in T(2)(M3(2)) be the metabelian 3‐group of nilpotency class c4 with 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. The cover of Gc,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

    where 0u. In particular, the cover is a finite set with +2 elements (+1 of them nontrivial), which are nonσ‐groups for even c4, and σ‐groups for odd c5.

  2. The Shafarevich cover of Gc,j(2) with respect to imaginary quadratic fields F is 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, if c5 is odd.

  3. The class‐c quotient with parameter k of the cover limit C(e) is isomorphic to a Schur σ‐group Qc(e,k)Sc,j(+3), for c=2+5 or to a nonσ‐group Qc(e,k)Gc,j(+2), for c=2+4. The precise correspondence between the parameters k and j is 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 class c=2+4, 0u.

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

  4. A parameterized family of fork topologies for second 3‐class groups Gal(F3(2)/F) of imaginary quadratic fields F is given uniformly for the states (ground state for =0, excited state for 1u) of transfer kernels with type E by the symmetric topology symbol

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

    with scaffold type c and the following invariants:

    distance d=4+2 (Definition 2.3), weighted distance w=5+3 (Definition 2.4),

    class increment Δcl=(2+5)(2+5)=0, coclass increment Δcc=(+3)2=+1,

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

Proof. We compare the uniform generator rank d1=2 of all involved groups Gc,j(r), c4, r2, 1j3, with their relation rank d2. Since d2=μ and the p‐multiplicator rank is μ=2 for Sc,j(r) with odd c=2+55 and r=+33, but μ=3 otherwise, 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=1 for Gc,j(r) with even c=2+4, r=+2, and child Tc+1,j(r), but ν=0 otherwise.

The execution of Algorithm 3.2 with input data p:=3, vb:=25, either i:=6, e:=0, or i:=8, e:=1, and cp:=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 subroutine IsAdmissible() 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.

Figure 3.

Projections Qc(e,k)Qc(e,k)/(Qc(e,k)) of the covers onto their metabelianizations.

Figure 4.

Branches of σ‐groups with complex type H.4 connected by the maintrunk.

Figure 5.

Tree topology of type E in the ground state.

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.

Figure 6.

Tree topology of type E in the first excited state.

Strictly speaking, the caption of Figure 3, in its full generality, is valid for e=1, M3(2)=243,8 only. 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.4 exclusively. We restrict the trees to σ‐groups indicated by green color. The top vertex 27,3 is intentionally drawn twice to avoid an overlap of the dense trees and to admit a uniform representation of periodic bifurcations.

Figure 7.

Tree topology of type E in the second excited state.

The tree with root 243,4 is not concerned by the quotients Qc(0,+1). It is sporadic and consists of periodically repeating finite saplings of depth 2 and 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 6561,614 and 6561,613#1;1#2;1 form 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)F and 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 towers of 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 qualitative topology problem for (m,n)=(1,2) is trivial, since the fork of Gp(1)F and Gp(2)F is simply the abelian root Gp(1)FClpF of the entire relevant descendant tree. However, the quantitative structure of the root path between Gp(2)F and Gp(1)F is 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)F with an arbitrary metabelian 3‐group M having abelianization M/M of type (3,3). Any attempt to determine the group G:=Gal(Fp()/F) of the p‐class tower Fp() of an algebraic number field F begins with a search for the metabelianization M:=G/G, i.e., the second derived quotient, of the p‐tower group G. M is also called the second p‐class group Gal(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=3 and a number field F with 3‐class group Cl3F of type (3,3), the structure of the root path from M to the root 9,2 is 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 s for a weighted edge of step size s1 with 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 paths P:=(πjM)0j of metabelian 3‐groups M with abelianization M/M of type (3,3), which are located on coclass trees. Let c denote the nilpotency class cl(M) and r the coclass cc(M) of M.

  1. If r=1 and c1, then P=X{1a}c1, where X{A,a,a}.

  2. If r=2 and c3, then either P=X{1b}c32a1a, where X{d,b,b}, or P=X{1c}c32a1a, where X{E,G,H,c}. An additional variant arises for r=2, c5, with P=X1X{1c}c42a1a, where X{G,H}.

  3. If r3 and cr+1, then either P=X{1b}c(r+1){2b}r22a1a, where X{d,b,b}, or P=X{1d}c(r+1){2b}r22a1a, where X{F,G,H,d}. An additional variant arises for r3, cr+3, with P=X1X{1d}c(r+2){2b}r22a1a, where X{G,H}.

In particular, the maximal possible step size is s=2, and the r1 edges with step size s=2 arise successively without gaps at the end of the path, except the trailing edge of step size s=1.

Proof. X always 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)9,2 with r=1 has a mainline of type a. Two of the coclass trees T(2)243,n with r=2, those with n{6,8}, have mainlines of type c and an additional scaffold of type a. For n=3, the mainline is of type b. The coclass trees T(r) with r3 behave uniformly with mainlines of type b or d and 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)j1 of a metabelian 3‐group M, observing that the root R=9,2 corresponds to the bicyclic quotient γ1/γ2(3,3) and the conspicuous trailing edge 1a corresponds 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+1 instead 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.

Corollary 3.1. (Sporadic root paths.)

As before, let M be a metabelian 3‐group with abelianization M/M(3,3), nilpotency class c:=cl(M), and coclass r:=cc(M). Assume that M is located outside of coclass trees.

  1. If r=2 and c=3, then P=X2a1a, where X{D,G,H}.

  2. If r=2 and c=4, then P=X1X2a1a, where X{G,H}.

  3. If r3 and c=r+1, then P=X{2b}r22a1a, where X{F,G,H}.

  4. If r3 and c=r+2, then P=X1X{2b}r22a1a, where X{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 subroutine IsAdmissible() which filters p‐groups G with 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 p and 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 type TKT, number nTotal of total kernels, number nFixed of fixed points, and further invariants occupation,repetitions,intersection describing the orbit of the TKT.

The output of Algorithm 3.3 is used for the subroutine IsAdmissible(G,p,t) in dependence on the parameter flag t. When the root R=243,8 is 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 sequences of finite p‐groups. Each of them has its benefits and drawbacks.

The bottom up strategy of 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 path in the descendant tree T(R). Its implementation in MAGMA is incredibly stable and robust without surpassing any internal limits up to logarithmic orders of 95 and even more. Only the consumption of CPU time becomes considerable in such extreme regions.

The top down strategy of expressing finite p‐groups as quotients of an infinite pro‐p group with given pro‐p presentation 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‐p presentation 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 G of 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 α and β depend 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.

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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 F were of type E.9 in its ground state, which admits two possibilities for the second 3‐class group, M2187,302 or 2187,306. Now we want to illustrate the way which led to the fork topologies in Theorem 3.5 by using the more convenient type E.8, where the group M is unique for every state, in particular, M2187,304 for the ground state.

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

Let F=Q(d) be an imaginary quadratic number field with 3‐class group Cl3F(3,3), and let E1,,E4 be the unramified cyclic cubic extensions of F.

Theorem 4.1. (First towers of type E.8.) Let the capitulation of 3‐classes of F in E1,,E4 be of type ϰ1F(1,2,3,1), which is called type E.8. Assume further that the 3‐class groups of E1,,E4 possess the abelian type invariants τ1F[T1,21,21,21], where T1{32,43,54}.

Then, the length of the 3‐class tower of F is precisely 3F=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=9,2 in the tree T(R). After two steps, 9,227,3243,8, we find the next root U5:=243,8 of 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;4 for the ground state T1=(32), M6561,2050#1;2 for the first excited state T1=(43), and M6561,2050(#1;1)2#1;2 for the second excited state T1=(54), where 2187,303=729,54#1;3 and 6561,2050=2187,303#1;1. The situation is visualized by Figure 2, where the three metabelianizations MG/G of 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=3 eliminates M as 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;4 with a siblings topology

E1 c 2E

which describes the relative location of M and 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=3 eliminates M and 6561,621#1;1#1;2 as candidates for the 3‐tower group G, according to Shafarevich, and we get the unique G6561,621#1;1#2;2 with 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;4 of the imaginary quadratic field F=Q(34867). The siblings topology is emphasized with red color, and the projection GMG/G is drawn in blue color.

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

Figure 7 shows the path to the 3‐tower group G729,54#2;3#1;1#2;1#1;1#2;2 of 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/G is drawn in blue color.

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5. Future developments

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

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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

  1. 1. Gamble G, Nickel W, O'Brien EA. ANU p‐Quotient — p‐Quotient and p‐Group Generation Algorithms. 2006. An accepted GAP package, available also in MAGMA
  2. 2. Mayer DC. Periodic bifurcations in descendant trees of finite p‐groups. Advances in Pure Mathematics. 2015;5(4):162–195. DOI: 10.4236/apm.2015.54020. Special Issue on Group Theory, March 2015
  3. 3. Besche HU, Eick B, O'Brien EA. The SmallGroups Library—a Library of Groups of Small Order. 2005. An accepted and refereed GAP package, available also in MAGMA
  4. 4. du Sautoy M. Counting p‐groups and nilpotent groups. Publications Mathématiques de l’Institut des Hautes Études Scientifiques. 2000;92:63–112
  5. 5. Eick B, Leedham‐Green C. On the classification of prime‐power groups by coclass. Bulletin of the London Mathematical Society. 2008;40(2):274–288
  6. 6. Mayer DC. New number fields with known p‐class tower. Tatra Mountains Mathematical Publications. 2015;64:21–57. DOI: 10.1515/tmmp‐2015‐0040. Special Issue on Number Theory and Cryptology 15
  7. 7. Newman MF. Determination of groups of prime‐power order. In: Group Theory, Canberra, 1975, Lecture Notes in Math. Vol. 573. Berlin: Springer; 1977. pp. 73–84
  8. 8. O'Brien EA. The p‐group generation algorithm. Journal of Symbolic Computation. 1990;9:677–698
  9. 9. Mayer DC. Artin transfer patterns on descendant trees of finite p‐groups. Advances in Pure Mathematics. 2016;6(2):66–104. DOI: 10.4236/apm.2016.62008. Special Issue on Group Theory Research, January 2016.
  10. 10. The MAGMA Group. MAGMA Computational Algebra System, Version 2.22‐7, Sydney. 2016. (Available from: http://magma.maths.usyd.edu.au)
  11. 11. Nebelung B. Klassifikation metabelscher 3‐Gruppen mit Faktorkommutatorgruppe vom Typ (3, 3) und Anwendung auf das Kapitulationsproblem [Inauguraldissertation]. Universität zu Köln; 1989
  12. 12. Mayer DC, Newman MF. Finite 3‐groups as Viewed from Class Field theory. Scotland, UK: Groups St. Andrews, Univ. of St. Andrews, Fife; 2013. Contributed presentation delivered on 11 August 2013. http://www.algebra.at/GroupsStAndrews2013.pdf
  13. 13. Mayer DC. Recent progress in determining p‐class field towers. Gulf Journal of Mathematics (Dubai, UAE). 2016;4(4):74–102. ISSN 2309‐4966
  14. 14. Shafarevich IR. Extensions with prescribed ramification points (Russian). Publications Mathématiques de l’Institut des Hautes Études Scientifiques.1964;18:71–95. (English transl. by J. W. S. Cassels in Amer. Math. Soc. Transl., II. Ser., 59 (1966), 128–149.)
  15. 15. Mayer DC. Periodic sequences of p‐class tower groups. Journal of Applied Mathematics and Physics. 2015;3(7):746–756. DOI: 10.4236/jamp.2015.37090. First International Conference on Groups and Algebras, 2015, Shanghai.
  16. 16. Mayer DC. Annihilator ideals of two‐generated metabelian p‐groups. Journal of Algebra and its Applications (JAA). (arXiv: 1603.09288v1 [math.GR] 30 Mar 2016.)
  17. 17. Mayer DC. Power‐commutator presentations for infinite sequences of 3‐groups, preprint. 2013. (Available from: https://www.researchgate.net/publication/256459683_Power‐commutator_presentations_for_infinite_sequences_of_3‐groups)
  18. 18. Scholz A, Taussky O. Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. J. Reine Angew. Math. 1934;171:19–41
  19. 19. Bush MR, Mayer DC. 3‐class field towers of exact length 3. Journal of Number Theory. 2015;147:766–777. DOI: 10.1016/j.jnt.2014.08.010
  20. 20. Mayer DC. Recent progress in determining p‐class field towers, 1st International Colloquium of Algebra, Number Theory, Cryptography and Information Security 2016, Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah, Fès, Morocco, invited keynote delivered on 12 November 2016. Available from: http://www.algebra.at/ANCI2016DCM.pdf
  21. 21. Ascione JA. On 3‐groups of second maximal class [Ph.D. Thesis]. Canberra: Austral. National Univ.; 1979
  22. 22. Ascione JA. On 3‐groups of second maximal class. Bulletin of the Australian Mathematical Society. 1980;21:473–474
  23. 23. Mayer DC, Newman MF. Finite 3‐groups with transfer kernel type F, in preparation. Available from: http://www.algebra.at/TransferSectionF.pdf

Written By

Daniel C. Mayer

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