_{1}

Recent examples of periodic bifurcations in descendant trees of finite p-groups with

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 [

The infinite families of Galois groups of p-class field towers with

For the specification of finite p-groups throughout this article, we use the identifiers of the SmallGroups database [

The first periodic bifurcations were discovered in August 2012 for the descendant trees of the 3-groups

candidates for Galois groups

p. 775). The result in [

Similar phenomena were found in May 2013 for the trees with roots

At the beginning of 2014, we investigated the root

In January 2015, we studied complex bicyclic biquadratic fields

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

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

Let p denote a prime number and suppose that G is a finite p-group or an infinite pro-p group with finite abelianization

In this situation, there exist

of intermediate normal subgroups

The following information is known [

Definition 3.1 By the Artin pattern of

onsisting of the multiple-layered TTT

If

As Emil Artin [

derived quotient

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

As our first application of periodic bifurcations in trees of 2-groups, we present a family of biquadratic number fields

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

Remark 4.1 Generally, the first layer of the transfer kernel type

Theorem 4.1 Let

Then the 2-class group

parameters

The Legendre symbol

•

of the coclass tree

•

More precisely, in the second case the following equivalences hold in dependence on the parameters

•

and varying

•

and varying

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

The ordered multi-layered transfer target type (TTT)

If we now denote by

Thus,

Proof. The underlying order of the 7 unramified quadratic, resp. bicyclic biquadratic, extensions of

For the TTT we use logarithmic abelian type invariants as explained in [

Concerning the TKT,

Proof. (Proof of Theorem 4.1)

Firstly, the equivalence

Next, we use the Artin pattern of

So far, we have been able to single out that

is used as input for a Magma program script [

Group

• CanIdentify Group() and Identify Group() if

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

• 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

Finally, the class and coclass of

With the aid of the computational algebra system MAGMA [

Recall that a pair

By means of the following invariants, the statistical distribution

The purely group theoretic pruned descendant tree was constructed in [

In

In

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.

Our second discovery of periodic bifurcations in trees of 3-groups will now be applied to a family of quadratic number fields

Theorem 6.1 Let

resp.

Further, let the integer

Then the 3-class field tower of

The metabelianization

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

The ordered multi-layered transfer target type (TTT)

If we now denote by

Thus,

Proof. First, we must establish the connection of the TTT of

Now, the statements are an immediate consequence of §§4.1-4.2 in our recent article [

Proof. (Proof of Theorem 6.1) First, we use the Artin pattern of

Now, Theorem 21.3 and Corollaries 21.2-21.3 in [

and

both with

With the aid of the computational algebra system MAGMA [

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

lianization

We have published this information in the Online Encyclopedia of Integer Sequences (OEIS) [

We emphasize that the results of section 6 provide the background for considerably stronger assertions than those made in [

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 [

Daniel C. Mayer, (2015) Periodic Sequences of p-Class Tower Groups. Journal of Applied Mathematics and Physics,03,746-756. doi: 10.4236/jamp.2015.37090