Periodic Sequences of p-Class Tower Groups

Recent examples of periodic bifurcations in descendant trees of finite p-groups with are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.

• p-Class Field Towers
• p-Principalization
• p-Class Groups
• Quadratic Fields
• Multiquadratic Fields
• Cubic Fields
• Finite p-Groups
• Parametrized Pc-Presentations
• p-Group Generation Algorithm

1. Introduction

In this article, we establish class field theoretic applications of the purely group theoretic discovery of periodic bifurcations in descendant trees of finite p-groups, as described in our previous papers [1] [21] [22] (pp. 182-193) and [2] (§6.2.2), and summarized in section §2.

The infinite families of Galois groups of p-class field towers with which are presented in sections §§4 and 6 can be divided into different kinds. Either they form infinite periodic sequences of uniform step size 1, and thus of fixed coclass. These are the classical and well-known coclass sequences whose virtual periodicity has been proved independently by du Sautoy and by Eick and Leedham-Green (see [1], §7, pp. 167-168). Or they arise from infinite paths of directed edges in descendant trees whose vertices reveal periodic bifurcations (see [1], Thm.21.1, p. 182, [1], Thm.21.3, p. 185, and [2], Thm.6.4). Extensive finite parts of the latter have been constructed computationally with the aid of the p-group generation algorithm by Newman [3] and O’Brien [4] (see [1] [12]-[18]), but their indefinitely repeating periodic pattern has not been proven rigorously, so far. They can be of uniform step size 2, as in §4, or of mixed alternating step sizes 1 and 2, as in §6, whence their coclass increases unboundedly.

2. Periodic Bifurcations in Trees of p-Groups

For the specification of finite p-groups throughout this article, we use the identifiers of the SmallGroups database [5] [6] and of the ANUPQ-package [7] implemented in the computational algebra systems GAP [8] and MAGMA [9]-[11], as discussed in [1] (§9, pp. 168-169).

The first periodic bifurcations were discovered in August 2012 for the descendant trees of the 3-groups and (see [1], §3, p. 163] and [1], Thm.21.3, p. 185), having abelian quotient in- variants, when we, in collaboration with Bush, conducted a search for Schur -groups as possible

candidates for Galois groups of three-stage towers of 3-class fields over complex quadratic base fields with and a certain type of 3-principalization [12] (Cor. 4.1.1,

p. 775). The result in [12] (Thm. 4.1, p. 774) will be generalized to more principalization types and groups of higher nilpotency class in section §6.

Similar phenomena were found in May 2013 for the trees with roots and of type but have not been published yet, since we first have to present a classification of all metabelian 3- groups with abelianization.

At the beginning of 2014, we investigated the root, which possesses an infinite balanced cover [2] (Dfn.6.1), and found periodic bifurcations in its decendant tree [2] (Thm.6.4).

In January 2015, we studied complex bicyclic biquadratic fields, called special Dirichlet fields by Hilbert [13], for whose 2-class tower groups presentations had been given by Azizi, Zekhnini

and Taous [14, Thm.2,(4)], provided the radicand d exhibits a certain prime factorization which ensures a 2- class group of type.

In Section §4, we use the viewpoint of descendant trees of finite metabelian 2-groups and our discovery of periodic bifurcations in the tree with root [1] (Thm.21.1, p. 182) to prove a group theoretic restatement of the main result in the paper [14], which connects pairs of positive integer parameters with vertices of the descendant tree by means of an injective mapping, as shown impressively in Figure 1.

3. Pattern Recognition via Artin Transfers

Let p denote a prime number and suppose that G is a finite p-group or an infinite pro-p group with finite abelianization of order with a positive integer exponent.

In this situation, there exist layers

of intermediate normal subgroups between G and its commutator subgroup. For each of them, we denote by the Artin transfer homomorphism from G to H [15]. In our recent papers [2] [3] [16], the components of the multiple-layered transfer target type (TTT) of G, resp. the multiple-layered transfer kernel type (TKT) of G, were defined by

The following information is known [16] to be crucial for identifying the metabelianization of a p-class tower group, but usually does not suffice to determine itself.

Definition 3.1 By the Artin pattern of we understand the pair

onsisting of the multiple-layered TTT and the multiple-layered TKT of.

If is the -tower group of a number field, then we put and speak about the Artin pattern of.

As Emil Artin [15] pointed out in 1929 already, using a more classical terminology, the concepts transfer target type (TTT) and transfer kernel type (TKT) of a base field, which we have now combined to the Artin pattern of, require a non-abelian setting of unramified extensions of. The reason is that the derived subgroup of an intermediate group between the p-tower group of and its commutator subgroup is an intermediate group between and the second derived subgroup. Therefore, the TTT of the p-tower group coincides with the TTT of any higher

derived quotient, for but not for, since,

Figure 1

Pairs of parameters distributed over

according to the isomorphism theorem. Similarly, we have the coincidence of TKTs, for.

4. Two-Stage Towers of 2-Class Fields

As our first application of periodic bifurcations in trees of 2-groups, we present a family of biquadratic number fields with 2-class group of type, discovered by Azizi, Zekhnini and Taous [14], whose 2-class tower groups are conjecturally distributed over infinitely many periodic coclass sequences, without gaps.

This claim is stronger than the statements in the following Theorem 4.1. The proof firstly consists of a group theoretic construction of all possible candidates for, identified by their Artin pattern, up to nilpotency class and coclass, thus having a maximal logarithmic order. (The first part is independent of the actual realization of the possible groups as 2-tower groups of suitable fields.) Secondly, evidence is provided of the realization of at least all those groups constructed in the first part whose logarithmic order does not exceed 11. The second part (see §5) is done by computing the Artin pattern of sufficiently many fields or by using more sophisticated ideas, presented in Theorem 4.1.

Remark 4.1 Generally, the first layer of the transfer kernel type of will turn out to be a permutation [1] (Dfn.21.1, p. 182) of the seven planes in the 3-dimensional -vector space. We are going to use the notation of [1] (Thm.21.1 and Cor.21.1).

Theorem 4.1 Let be a complex bicyclic biquadratic Dirichlet field with radicand, where, and are prime numbers such that and.

Then the 2-class group of is of type, the 2-class field tower of is metabelian (with exactly two stages), and the isomorphism type of the Galois group of the maximal unramified pro-2 extension of is characterized uniquely by the pair of positive integer

parameters defined by the 2-class numbers and of the complex quadratic fields and.

The Legendre symbol decides whether is a descendant of or:

• the first layer TKT is a permutation with five fixed points and a single 2-cycle belongs to the mainline

of the coclass tree.

• the first layer TKT is a permutation with a single fixed point and three 2-cycles is a descendant of the group, that is.

More precisely, in the second case the following equivalences hold in dependence on the parameters, where denotes a foregiven upper bound:

• (with fixed) belongs to the mainline

and varying, of the coclass tree.

• (with fixed) belongs to the unique periodic coclass sequence

and varying, whose members possess a permutation as their first layer transfer kernel type, of the coclass tree.

We add a corollary which gives the Artin pattern of the groups in Theorem 4.1, firstly, since it is interesting in its own right, and secondly, because we are going to use its proof as a starting point for the proof of Theorem 4.1.

Corollary 4.1 Under the assumptions of Theorem 4.1, the Artin pattern of the 2- tower group of the biquadratic field is given as follows:

The ordered multi-layered transfer target type (TTT) of the Galois group is given by, , and

If we now denote by, , the norm class groups of the seven unramified quadratic extensions, then the ordered multi-layered transfer kernel type (TKT) of the Galois group is given by, , , and

Thus, is always a permutation of the norm class groups. For it contains five fixed points and a single 2-cycle, and otherwise it contains a single fixed point and three 2-cycles.

Proof. The underlying order of the 7 unramified quadratic, resp. bicyclic biquadratic, extensions of is taken from [14] (§2.1, Thm.1, (3), (5)).

For the TTT we use logarithmic abelian type invariants as explained in [2] (§2). is taken from [14] (§2.2, Thm.2, (1)), from [14] (2.3, Thm.3, (1), (2)), and from [14] (§2.2, Thm.2, (5)).

Concerning the TKT, is trivial, are taken from [14] (§2.3, Thm.3, (3)-(5)), and is total, due to the Hilbert/Artin/Furtwängler principal ideal theorem.

Proof. (Proof of Theorem 4.1)

Firstly, the equivalence is proved in [14] (3, Lem.5).

Next, we use the Artin pattern of, as given in Corollary 4.1, to narrow down the possibilities for. The possible class-2 quotients of are exactly the immediate descendants of the root, that is, three vertices of step size 1, nine vertices of step size 2, and ten vertices of step size 3. Among all descendants of, the mainline vertices of the tree are characterized uniquely by the fact that their first layer TKT is a permutation with five fixed points and a single 2-cycle, and that their first layer TTT contains the unique polarized (i.e. parameter dependent) component. Note that the mainline vertices of the tree reveal the same six stable (i.e. parameter independent) components of the accumulated (unordered) first layer TTT, but their first layer TKT contains three 2-cycles, similarly as for descendants of. However, vertices of the complete descendant tree are characterized uniquely by six stable components of their first layer TTT.

So far, we have been able to single out that must be a descendant of either or, by means of Artin patterns, without knowing a presentation. Now, the parametrized presentation for the group in [14] (§2.2, Thm.2, (4)),

is used as input for a Magma program script [10] [11] which identifies a 2-group, given by generators and relations,

Group, with the aid of the following functions:

• CanIdentify Group() and Identify Group() if,

• Is In Small Group Database(), pQuotient(), Number Of Small Groups(), Small Group() and Is Isomorphic() if, and

• Generatep Groups(), resp. a recursive call of Descendants() (using Nuclear Rank() for the recursion), and Is Isomorphic() if.

The output of the Magma script is in perfect accordance with the pruned descendant tree, as described in Theorem 21.1 and Corollary 21.1 of [1] (pp.182-183).

Finally, the class and coclass of are given in [14] (§2.2, Thm.2, (6)).

5. Computational Results for Two-Stage Towers

With the aid of the computational algebra system MAGMA [11], we have determined the pairs of parameters, investigated in [14], for all 11753 square free radicands of the shape in Theorem 4.1 which occur in the range. As mentioned at the beginning of §4, the result supports the conjecture that the corresponding 2-tower groups cover the pruned tree without gaps.

Recall that a pair contains information on the 2-class numbers of complex quadratic fields. So we have a reduction of hard problems for biquadratic fields to easy questions about quadratic fields.

By means of the following invariants, the statistical distribution of parameter pairs is visualized on the pruned descendant tree, using the injective (and probably even bijective) mapping. For each fixed individual pair, we define its minimal radicand as an absolute invariant:

The purely group theoretic pruned descendant tree was constructed in [1] (§21.1, pp. 182-184), and was shown in [1] (§10.4.1, Figure 7, p. 175), with vertices labelled by the standard identifiers in the SmallGroups Library [5] [6] or of the ANUPQ-package [7].

In Figure 1, a pair of parameters is placed adjacent to the corresponding vertex of the pruned descendant tree. There are no overlaps, since the mapping is injective. Each vertex is additionally labelled with a formal identifier, as used in [1] (Cor.21.1).

In Figure 2, the minimal radicand for which the adjacent vertex is realized as the corresponding group, is shown underlined and with boldface font.

Vertices within the support of the distribution are surrounded by an oval. The oval is drawn in horizontal orientation for mainline vertices and in vertical orientation for vertices in other periodic coclass sequences.

6. Three-Stage Towers of 3-Class Fields

Our second discovery of periodic bifurcations in trees of 3-groups will now be applied to a family of quadratic number fields with 3-class group of type, originally investigated by ourselves in [16]-[18], and extended by Boston, Bush and Hajir in [19]. The 3-class tower groups of these fields are conjecturally distributed over six periodic sequences arising from repeated bifurcations (of the new kind which was unknown up to now), whereas it is proven that their metabelianizations populate six well-known periodic coclass sequences of fixed coclass 2.

Theorem 6.1 Let be a complex quadratic field with discriminant, having a 3-class group of type, such that its 3-principalization in the four unramified cyclic cubic extensions is given by one of the following two first layer TKTs

resp.

Further, let the integer denote a foregiven upper bound.

Then the 3-class field tower of is non-metabelian with exactly three stages, and the isomorphism type of the Galois group of the maximal unramified pro-3 extension of is characterized uniquely by the positive integer parameter defined by the 3-class number of the simply real non-Galois cubic subfield of the distinguished polarized extension among (i.e., , resp.):

Figure 2

Minimal radicands distributed over

The metabelianization of the Schur -group, that is the Galois group of the maximal metabelian unramified 3-extension of is unbalanced and given by

Again, we first state a corollary whose proof will initialize the proof of Theorem 6.1.

Corollary 6.1 Under the assumptions of Theorem 6.1, the Artin pattern of the 3- tower group of the complex quadratic field is given as follows:

The ordered multi-layered transfer target type (TTT) of the Galois group is given by, , and

If we now denote by, , the norm class groups of the four unramified cyclic cubic extensions, then the ordered multi-layered transfer kernel type (TKT) of the Galois group is given by, , and

Thus, is not a permutation of the norm class groups. For it contains a single or no fixed point and no 2-cycle, and for it contains three or two fixed points and no 2-cycle.

Proof. First, we must establish the connection of the TTT of with the distinguished non-Galois simply real cubic field. Anticipating the partial result of Theorem 6.1 that the metabelianization of must be of coclass, we can determine the 3-class numbers of all four non-Galois cubic subfields with the aid of Theorem 4.2 in [17] (p. 489): with respect to the normalization in this theorem, we have

and uniformly for, since, which implies, and has no defect of commutativity. The parameter is the index of nilpotency of, whence the nilpotency class is given by.

Now, the statements are an immediate consequence of §§4.1-4.2 in our recent article [2], where the claims are reduced to theorems in our earlier papers: [16] (Thm.1.3, p. 405), and, more generally, [18] (Thm.4.4, p.440 and Tbl.4.7, p. 441). We must only take into consideration that the 3-class group of is nearly homocyclic with abelian type invariants, since, and thus.

Proof. (Proof of Theorem 6.1) First, we use the Artin pattern of, as given in Corollary 6.1, to narrow down the possibilities for. The possible class-3 quotients of are exactly the immediate descendants of the common class-2 quotient of all 3-groups with abelianization of type (apart from), that is, four vertices of step size 1 [1] (Figure 3), and seven vertices of step size 2 [1] (Figure 4). All descendants of the former are of coclass 1 and reveal the same three stable (i.e. parameter independent) components of the first layer TTT, according to [2] (Thm.3.2, (1)), which does not agree with the required TTT of. Among the latter, the criterion [12] (Cor.3.0.2, p. 772) rejects three of the seven vertices, , since the TKT of does not contain a 2-cycle, and are dis- couraged, since they are terminal. The remaining two vertices are exactly the parents of the decisive groups, where periodic bifurcations set in.

Now, Theorem 21.3 and Corollaries 21.2-21.3 in [1] (pp. 185-187) show that, using the local notation of Corollary 21.2,

and

both with.

7. Computational Results for Three-Stage Towers

With the aid of the computational algebra system MAGMA [11], where the class field theoretic techniques of Fieker [20] are implemented, we have determined the Artin pattern of all complex quadratic fields with discriminants in the range, whose first layer TTT had been

Figure 3

Minimal absolute discriminants distributed over

precomputed by Boston, Bush and Hajir in the database underlying the numerical results in [19].

Figure 3, resp. 4, shows the minimal absolute discriminant, underlined and with boldface font, for which the adjacent vertex of the coclass tree, resp., is realized as the metabe-

Figure 4

Minimal absolute discriminants distributed over

lianization of the 3-tower group of. Vertices within the support of the distribution are surrounded by an oval. The corresponding projections have been visualized in the Figure 8 and Figure 9 of [1] (pp. 188-189).

We have published this information in the Online Encyclopedia of Integer Sequences (OEIS) [21], sequences A247692 to A247697.

We emphasize that the results of section 6 provide the background for considerably stronger assertions than those made in [12]. Firstly, since they concern four TKTs E.6, E.14, E.8, E.9 instead of just TKT E.9 [2] (§4), and secondly, since they apply to varying odd nilpotency class instead of just class 5.

Acknowledgements

We gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008-N25. We are indebted to Nigel Boston, Michael R. Bush and Farshid Hajir for kindly making available an unpublish- ed database containing numerical results of their paper [19].

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