“Artin transfer (group theory)”的意思、由来-开放百科全书

from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index.

Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms

a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with

parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees.

Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups

and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers.

These strategies of pattern recognition are useful in purely group theoretic context,

as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers.

Transversals of a subgroup

Of course, this element with subscript

which represents the principal coset (i.e., the subgroup

itself)

We must check whether the image of this transversal under the homomorphism is again a transversal.

By assumption, we have the disjoint left coset decomposition

which comprises two statements simultaneously.

To show that the images of the cosets remain disjoint we need the property

of the homomorphism

Permutation representation

we have the following relations between the monomials and permutations corresponding to an element

Artin transfer

Independence of the transversal

Finally, due to the commutativity of the quotient group

and the fact that

and

are permutations,

It is clear that a similar proof shows that the Artin transfer is independent of the choice between two different right transversals.

It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal.

For this purpose, we select the special right transversal

associated to the left transversal

The last step is justified by the fact that the Artin transfer is a homomorphism.

Artin transfers as homomorphisms

It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial representation. The images of the factors

are given by

which is a very peculiar law of composition discussed in more detail in the following section.

Wreath product of H and S(n)

can also be interpreted by endowing the cartesian product

with a special law of composition

For

, the wreath product of the associated monomials and permutations is given by

In fact, it is an injective homomorphism, also called a monomorphism or embedding,

which cannot be injective if

is infinite or at least of an order bigger than

, the factorial.

For a homomorphism to be injective, it suffices to show the triviality of its kernel.

Finally, an application of the inverse inner automorphism with

yields

, as required for injectivity.

Whereas Huppert^[1] uses the monomial representation for defining the Artin transfer,

and to merely illustrate the homomorphism property of the Artin transfer with the aid of the monomial representation.

Composition of Artin transfers

Finally, we want to emphasize the structural peculiarity of the monomial representation

whereas the action on the first component

is independent of the second component

Wreath product of S(m) and S(n)

The permutations

, which arose as second components of the monomial representation

Using the peculiarities of the composition of Artin transfers in the previous section,

for the Fréchet derivative in

of the compositum of differentiable functions

The above considerations establish a third representation, the stabilizer representation,

As opposed to the latter, the stabilizer representation cannot be injective, in general.

Cycle decomposition

The reason for this fact is that we obtain another left transversal of

by putting

The cycle decomposition corresponds to a double coset decomposition

of the group

modulo the cyclic group

and modulo the subgroup

. It was actually this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.^[3]

Transfer to a normal subgroup

Hence, the formula for the image of

under the Artin transfer

in the previous section takes the particular shape

In particular, the inner transfer of an element

is given as a symbolic power

The other extreme is the outer transfer of an element

which generates

modulo

Explicit implementations of Artin transfers in the simplest situations are presented in the following section.

Computational implementation

Abelianization of type (p,p)

For each

, let

be the Artin transfer homomorphism from

to the abelianization of

according to equation (16) in the last Corollary, which can also be expressed by a product of two pth powers,

It should be pointed out that the complete specification of the Artin transfers

also requires explicit knowledge of the derived subgroups

Abelianization of type (p²,p)

However, for the distinguished maximal subgroup

, for which the factor group

is bicyclic of type

Again, it should be emphasized that the structure of the derived subgroups

and

must be known

Transfer kernels and targets

Let

be a group with finite abelianization

. Suppose that

denotes the family of all subgroups

which contain the commutator subgroup

and are therefore necessarily normal, enumerated by means of the finite index set

. For each

, let

be the Artin transfer from

to the abelianization

The family of normal subgroups

is called the transfer kernel type (TKT) of

with respect to

, and the family of abelianizations (resp. their abelian type invariants)

is called the transfer target type (TTT) of

with respect to

Both families are also called multiplets whereas a single component will be referred to as a singulet.

Important examples for these concepts are provided in the following two sections.

Abelianization of type (p,p)

Let

be a p-group with abelianization

of elementary abelian type

. Then

has

maximal subgroups

of index

. For each

, let

be the Artin transfer homomorphism from

to the abelianization of

The family of normal subgroups

is called the transfer kernel type (TKT) of

with respect to

The orbit

of any representative

is an invariant of the p-group

and is called its transfer kernel type, briefly TKT.

Let

denote the counter of total transfer kernels

, which is an invariant of the group

However, for

, there do not exist non-abelian

-groups

with

, which must be metabelian of maximal class, such that

. Only the elementary abelian

-group

has

. See Figure 5.

In the following concrete examples for the counters

, and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by H. U. Besche, B. Eick and E. A. O'Brien

Abelianization of type (p²,p)

Let

be a p-group with abelianization

of non-elementary abelian type

. Then

possesses

maximal subgroups

of index

, and

subgroups

of index

Suppose that

is the distinguished maximal subgroup which is the product of all subgroups of index

, and

is the distinguished subgroup of index

which is the intersection of all maximal subgroups, that is the Frattini subgroup

First layer

For each

, let

be the Artin transfer homomorphism from

to the abelianization of

The family

is called the first layer transfer kernel type of

with respect to

and

, and is identified with

, where

Here, we observe that each first layer transfer kernel is of exponent

with respect to

and consequently cannot coincide with

for any

, since

is cyclic of order

, whereas

is bicyclic of type

Second layer

For each

, let

be the Artin transfer homomorphism from

to the abelianization of

The family

is called the second layer transfer kernel type of

with respect to

and

, and is identified with

, where

Transfer kernel type

Combining the information on the two layers, we obtain the (complete) transfer kernel type

of the p-group

with respect to

and

The distinguished subgroups

and

are unique invariants of

and should not be renumerated. However, independent renumerations of the remaining maximal subgroups

and the transfers

by means of a permutation

, and of the remaining subgroups

of index

and the transfers

by means of a permutation

, give rise to new TKTs

with respect to

and

, identified with

, where

and

with respect to

and

, identified with

, where

It is adequate to view the TKTs

and

as equivalent. Since we have

, resp.

, the relations between

and

, resp.

and

, are given by

, resp.

. Therefore,

is another representative of the orbit

under the operation

of the product of two symmetric groups

on the set of all pairs of mappings from

, where the extensions

and

of a permutation

are defined by

and

, and formally

, and

The orbit

of any representative

is an invariant of the p-group

and is called its transfer kernel type, briefly TKT.

Connections between layers

The Artin transfer

from

to a subgroup

of index

(

) is the compositum

of the induced transfer

from

and the Artin transfer

from

, for any intermediate subgroup

of index

(

). There occur two situations:

This causes restrictions for the transfer kernel type

of the second layer, since

, and thus

Furthermore, when

with

and

, an element

(

) which is of order

with respect to

, can belong to the transfer kernel

only if its

th power

is contained in

, for all intermediate subgroups

, and thus:

Inheritance from quotients

The common feature of all parent-descendant relations between finite p-groups is that the parent

is a quotient

of the descendant

by a suitable normal subgroup

. Thus, an equivalent definition can be given by selecting an epimorphism

from

onto a group

whose kernel

plays the role of the normal subgroup

. Then the group

can be viewed as the parent of the descendant

. In the following sections, this point of view will be taken, generally for arbitrary groups, not only for finite p-groups.

Passing through the abelianization

is a homomorphism from a group

to an abelian group

, then there exists a unique homomorphism

such that

, where

denotes the canonical projection onto the abelianization

. The kernel of

is given by

. The situation is visualized in Figure 1.

This statement is a consequence of the second Corollary in the article on the induced homomorphism. Nevertheless, we give an independent proof for the present situation.

The uniqueness of

is a consequence of the condition

, which implies that

must be defined by

, for any

For the commutator of

, we have

, since

is abelian. Thus, the commutator subgroup

is contained in the kernel

, and this finally shows that the definition of

is independent of the coset representative,

TTT singulets

Let

and

be groups such that

is the image of

under an epimorphism

and

is the image of a subgroup

The commutator subgroup of

is the image of the commutator subgroup of

, that is

Therefore,

induces a unique epimorphism

, and thus

is an epimorphic image of

, or with other words, a quotient of

Moreover, if

, then the map

is an isomorphism, and the abelianizations

are isomorphic.

This claim is a consequence of the Main Theorem in the article on the induced homomorphism. Nevertheless, an independent proof is given as follows.

Secondly, the epimorphism

can be restricted to an epimorphism

. According to the previous section, the composite epimorphism

from

onto the abelian group

factors through

by means of a uniquely determined epimorphism

such that

Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants by putting

TKT singulets

Suppose that

and

are groups,

is the image of

under an epimorphism

, and

is the image of a subgroup

of finite index

. Let

be the Artin transfer from

and

be the Artin transfer from

, then the image

of a left transversal

is a left transversal of

, and the inclusion

holds.

Let

be a left transversal of

. Then

is a disjoint union but

is not necessarily disjoint. For

, we have

for some element

. However, if the condition

is satisfied, then we are able to conclude that

, and thus

This has been shown already in the Proposition of the initial section on transversals of a subgroup.

Let

be the epimorphism obtained in the manner indicated in the previous section.

For the image of

under the Artin transfer, we have

. Since

, the right hand side equals

, provided that

is a left transversal of

, which is correct, when

. This shows that the diagram in Figure 3 is commutative, that is

Finally, if

, then the previous section has shown that

is an isomorphism. Using the inverse isomorphism, we get

, which proves

In view of the results in the present section, we are able to define a partial order of transfer kernels by setting

TTT and TKT multiplets

Suppose

and

are groups,

is the image of

under an epimorphism

, and both groups have isomorphic finite abelianizations

Let

denote the family of all subgroups

which contain the commutator subgroup

(and thus are necessarily normal), enumerated by means of the finite index set

, and let

be the image of

under

, for each

Assume that, for each

denotes the Artin transfer from

to the abelianization

, and

denotes the Artin transfer from

to the abelianization

, called the (partial) transfer kernel type (TKT) of

with respect to

, and

Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws:

Inherited automorphisms

A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applications to descendant trees.

Let

and

be groups such that

is the image of

under an epimorphism

. Suppose that

is an automorphism of

, then there exists a unique epimorphism

such that

. If

, then

is also an automorphism.

The justification for these facts is based on the isomorphic representation

, which permits to identify

for all

and proves the uniqueness of

. If

, then the consistency follows from

. And if

, then injectivity of

is a consequence of

, since

The reason for the injectivity of

is that

, since

is a characteristic subgroup of

is called a σ-group, if there exists an automorphism

such that the induced automorphism acts like the inversion on

, that is,

, resp.

, for all

The supplementary inheritance property asserts that, if

is a

-group and

, then

is also a

-group, the required automorphism being

This can be seen by applying the epimorphism

to the equation

, for

, which yields

, for all

Stabilization criteria

In this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section are applied to the simplest case, which is characterized by the following

The parent

of a group

is the quotient

by the last non-trivial term

of the lower central series of

, where

denotes the nilpotency class of

. The corresponding epimorphism

from

onto

is the canonical projection, whose kernel is given by

the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite p-groups.

Let

be a prime number. Suppose that

is a non-abelian finite p-group of nilpotency class

. Then the TTT and the TKT of

and of its parent

are comparable in the sense that

and

the investigated groups are supposed to be finite metabelian p-groups

with elementary abelianization

of rank

, that is of type

A metabelian p-group

of coclass

and of nilpotency class

shares the last

components of the TTT

and of the TKT

with its parent

The condition

is indeed necessary for the partial stabilization criterion. For odd primes

, the extra special

-group

of order

and exponent

has nilpotency class

only, and the last

components of its TKT

are strictly smaller than the corresponding components of the TKT

of its parent

which is the elementary abelian

-group of type

For

, both extra special

-groups of coclass

and class

, the ordinary quaternion group

with TKT

and the dihedral group

with TKT

, have strictly smaller last two components of their TKTs than their common parent

with TKT

A metabelian p-group

of coclass

and of nilpotency class

, that is, with index of nilpotency

, shares all

components of the TTT

and of the TKT

with its parent

, provided it has positive defect of commutativity

The condition

is indeed necessary for total stabilization. To see this it suffices to consider the first component of the TKT only. For each nilpotency class

, there exist (at least) two groups

with TKT

and

with TKT

, both with defect

, where the first component of their TKT is strictly smaller than the first component of the TKT

of their common parent

A metabelian 3-group

with abelianization

, coclass

and nilpotency class

shares the last two (among the four) components of the TTT

and of the TKT

with its parent

The condition

is indeed unavoidable for partial stabilization, since there exist several

-groups of class

, for instance those with SmallGroups identifiers

, such that the last two components of their TKTs

are strictly smaller than the last two components of the TKT

of their common parent

A metabelian 3-group

with abelianization

, coclass

, nilpotency class

and cyclic centre

shares all four components of the TTT

and of the TKT

with its parent

The condition of a cyclic centre is indeed necessary for total stabilization, since for a group with bicyclic centre there occur two possibilities.

Summarizing, we can say that the last four criteria underpin the fact that Artin transfers provide a marvellous tool for classifying finite p-groups.

In the following sections, it will be shown how these ideas can be applied for endowing descendant trees with additional structure, and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in pure group theory and in algebraic number theory.

Structured descendant trees (SDTs)

This section uses the terminology of descendant trees in the theory of finite p-groups.

In Figure 4, a descendant tree with modest complexity is selected exemplarily to demonstrate how Artin transfers provide additional structure for each vertex of the tree.

More precisely, the underlying prime is

, and the chosen descendant tree is actually a coclass tree having a unique infinite mainline, branches of depth

, and strict periodicity of length

setting in with branch

Branches

and

form the primitive period such that

, for odd

, and

, for even

The root of the tree is the metabelian

-group with identifier

, that is, a group of order

and with counting number

. It should be emphasized that this root is not coclass settled, whence its entire descendant tree

is of considerably higher complexity than the coclass-

subtree

, whose first six branches are drawn in the diagram of Figure 4.

The additional structure can be viewed as a sort of coordinate system in which the tree is embedded. The horizontal abscissa is labelled with the transfer kernel type (TKT)

, and the vertical ordinate is labelled with a single component

of the transfer target type (TTT). The vertices of the tree are drawn in such a manner that members of periodic infinite sequences form a vertical column sharing a common TKT. On the other hand, metabelian groups of a fixed order, represented by vertices of depth at most

, form a horizontal row sharing a common first component of the TTT. (To discourage any incorrect interpretations, we explicitly point out that the first component of the TTT of non-metabelian groups or metabelian groups, represented by vertices of depth

, is usually smaller than expected, due to stabilization phenomena!) The TTT of all groups in this tree represented by a big full disk, which indicates a bicyclic centre of type

, is given by

with varying first component

, the nearly homocyclic abelian

-group of order

, and fixed further components

and

, where the abelian type invariants are either written as orders of cyclic components or as their

-logarithms with exponents indicating iteration. (The latter notation is employed in Figure 4.) Since the coclass of all groups in this tree is

, the connection between the order

and the nilpotency class is given by

Pattern recognition

For searching a particular group in a descendant tree by looking for patterns defined by the kernels and targets of Artin transfers, it is frequently adequate to reduce the number of vertices in the branches of a dense tree with high complexity by sifting groups with desired special properties, for example

The result of such a sieving procedure is called a pruned descendant tree with respect to the desired set of properties.

However, in any case, it should be avoided that the main line of a coclass tree is eliminated, since the result would be a disconnected infinite set of finite graphs instead of a tree.

For example, it is neither recommended to eliminate all

-groups in Figure 4 nor to eliminate all groups with TKT

In Figure 4, the big double contour rectangle surrounds the pruned coclass tree

, where the numerous vertices with TKT

are completely eliminated. This would, for instance, be useful for searching a

-group with TKT

and first component

of the TTT. In this case, the search result would even be a unique group. We expand this idea further in the following detailed discussion of an important example.

Historical example

The oldest example of searching for a finite p-group by the strategy of pattern recognition via Artin transfers goes back to 1934, when A. Scholz and O. Taussky

tried to determine the Galois group

of the Hilbert

-class field tower, that is the maximal unramified pro-

extension

, of the complex quadratic number field

They actually succeeded in finding the maximal metabelian quotient

, that is the Galois group of the second Hilbert

-class field

However, it needed

years until M. R. Bush and D. C. Mayer, in 2012, provided the first rigorous proof

that the (potentially infinite)

-tower group

coincides with the finite

-group

of derived length

, and thus the

-tower of

has exactly three stages, stopping at the third Hilbert

-class field

The search is performed with the aid of the p-group generation algorithm by M. F. Newman

For the initialization of the algorithm, two basic invariants must be determined. Firstly, the generator rank

of the p-groups to be constructed. Here, we have

and

is given by the

-class rank of the quadratic field

. Secondly, the abelian type invariants of the

-class group

. These two invariants indicate the root of the descendant tree which will be constructed successively. Although the p-group generation algorithm is designed to use the parent-descendant definition by means of the lower exponent-p central series, it can be fitted to the definition with the aid of the usual lower central series. In the case of an elementary abelian p-group as root, the difference is not very big. So we have to start with the elementary abelian

-group of rank two, which has the SmallGroups identifier

, and to construct the descendant tree

. We do that by iterating the p-group generation algorithm, taking suitable capable descendants of the previous root as the next root, always executing an increment of the nilpotency class by a unit.

As explained at the beginning of the section Pattern recognition, we must prune the descendant tree with respect to the invariants TKT and TTT of the

-tower group

, which are determined by the arithmetic of the field

(exactly two fixed points and no transposition) and

Further, any quotient of

must be a

-group, enforced by number theoretic requirements for the quadratic field

The root

has only a single capable descendant

of type

. In terms of the nilpotency class,

is the class-

quotient

and

is the class-

quotient

. Since the latter has nuclear rank two, there occurs a bifurcation

, where the former component

can be eliminated by the stabilization criterion

for the TKT of all

-groups of maximal class.

Due to the inheritance property of TKTs, only the single capable descendant

qualifies as the class-

quotient

There is only a single capable

-group

among the descendants of

. It is the class-

quotient

and has nuclear rank two.

This causes the essential bifurcation

in two subtrees belonging to different coclass graphs

and

. The former contains the metabelian quotient

with two possibilities

, which are not balanced with relation rank

bigger than the generator rank. The latter consists entirely of non-metabelian groups and yields the desired

-tower group

as one among the two Schur

-groups

and

with

where, for each of the possible successive p-quotients

of the

-tower group

, the nilpotency class is denoted by

, the nuclear rank by

, and the p-multiplicator rank by

Commutator calculus

This section shows exemplarily how commutator calculus can be used for determining the kernels and targets of Artin transfers explicitly.

As a concrete example we take the metabelian

-groups with bicyclic centre, which are represented by big full disks as vertices, of the coclass tree diagram in Figure 4.

They form ten periodic infinite sequences, four, resp. six, for even, resp. odd, nilpotency class

, and can be characterized with the aid of a parametrized polycyclic power-commutator presentation:

The transfer target type (TTT) of the group

depends only on the nilpotency class

, is independent of the parameters

, and is given uniformly by

This phenomenon is called a polarization, more precisely a uni-polarization,

The transfer kernel type (TKT) of the group

is independent of the nilpotency class

, but depends on the parameters

, and is given by

For even nilpotency class, the two groups of types H.4 and E.14, which differ in the sign of the parameter

only, are isomorphic.

As a preparation, it is useful to compile a list of some commutator relations, starting with those given in the presentation,

The maximal subgroups of

are taken in a similar way as in the section on the computational implementation, namely

Their derived subgroups are crucial for the behavior of the Artin transfers. By making use of the general formula

, where

As the first main result, we are now in the position to determine the abelian type invariants of the derived quotients:

Now we come to the kernels of the Artin transfer homomorphisms

. It suffices to investigate the induced transfers

and to begin by finding expressions for the images

of elements

, which can be expressed in the form

with exponents

. First, we exploit outer transfers as much as possible:

Next, we treat the unavoidable inner transfers, which are more intricate. For this purpose, we use the polynomial identity

to obtain:

The following equivalences, for any

, finish the justification of the statements:

Consequently, the last three components of the TKT are independent of the parameters

, which means that both, the TTT and the TKT, reveal a uni-polarization at the first component.

Systematic library of SDTs

The aim of this section is to present a collection of structured coclass trees (SCTs) of finite p-groups with parametrized presentations and a succinct summary of invariants.

The trees are arranged according to increasing coclass

and different abelianizations within each coclass.

To keep the descendant numbers manageable, the trees are pruned by eliminating vertices of depth bigger than one.

Further, we omit trees where stabilization criteria enforce a common TKT of all vertices, since we do not consider such trees as structured any more.

We refrain from giving justifications for invariants, since the way how invariants are derived from presentations was demonstrated exemplarily in the section on commutator calculus

Coclass 1

For each prime

, the unique tree of p-groups of maximal class is endowed with information on TTTs and TKTs, that is,

for

, and

for

. In the last case, the tree is restricted to metabelian

-groups.

The

-groups of coclass

in Figure 5 can be defined by the following parametrized polycyclic pc-presentation, quite different from Blackburn's presentation.

The branches are strictly periodic with pre-period

and period length

, and have depth

and width

Polarization occurs for the third component and the TTT is

, only dependent on

and with cyclic

There are two exceptions, the abelian root with

and

, and the usual quaternion group with

and

The

-groups of coclass

in Figure 6 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Blackburn's presentation.

The branches are strictly periodic with pre-period

and period length

, and have depth

and width

Polarization occurs for the first component and the TTT is

, only dependent on

and

There exist three exceptions, the abelian root with

, the extra special group of exponent

with

and

, and the Sylow

-subgroup of the alternating group

with

The metabelian

-groups of coclass

in Figure 7 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Miech's presentation.

The (metabelian!) branches are strictly periodic with pre-period

and period length

, and have depth

and width

. (The branches of the complete tree, including non-metabelian groups, are only virtually periodic and have bounded width but unbounded depth!)

Polarization occurs for the first component and the TTT is

, only dependent on

and the defect of commutativity

There exist three exceptions, the abelian root with

, the extra special group of exponent

with

and

, and the group

with

Coclass 2

Abelianization of type (p,p)

Three coclass trees,

and

for

, are endowed with information concerning TTTs and TKTs.

On the tree

, the

-groups of coclass

with bicyclic centre in Figure 8 can be defined by the following parametrized polycyclic pc-presentation.

The branches are strictly periodic with pre-period

and period length

, and have depth

and width

Polarization occurs for the first component and the TTT is

, only dependent on

On the tree

, the

-groups of coclass

with bicyclic centre in Figure 9 can be defined by the following parametrized polycyclic pc-presentation.

The branches are strictly periodic with pre-period

and period length

, and have depth

and width

Polarization occurs for the second component and the TTT is

, only dependent on

Abelianization of type (p²,p)

Abelianization of type (p,p,p)

Coclass 3

Abelianization of type (p²,p)

Abelianization of type (p,p,p)

Arithmetical applications

frequently permits to draw conclusions about the entire p-class field tower of

Example

To be specific, let

and consider complex quadratic fields

with fixed signature

surrounding a realized vertex

with a circle and attaching an adjacent underlined boldface integer

Then we obtain the arithmetically structured coclass trees (ASCTs) in Figures 10 and 11,

which, in particular, give an impression of the actual distribution of second

-class groups.

Concerning the periodicity of occurrences of second

-class groups

of complex quadratic fields,

that only every other branch of the trees in Figures 10 and 11 can be populated by these metabelian

-groups and that

the ground state and three excited states are known for each of the six sequences,

The minimal absolute discriminants of the various states of each of the six periodic sequences are presented in Table 2.

Data for the ground states (GS) and the first excited states (ES1) has been taken from D. C. Mayer,

most recent information on the second, third and fourth excited states (ES2, ES3, ES4) is due to N. Boston, M. R. Bush and F. Hajir.

In contrast, let us secondly select the sporadic part

of the coclass graph

for demonstrating that

another way of attaching additional arithmetical structure to descendant trees is to display the counter

of fields with absolute discriminants below a given upper bound

, for instance

this gives the relative frequency as an approximation to the asymptotic density of the population

Comparison of various primes

The resulting structured sporadic coclass graph is shown in Figure 13 for

, in Figure 14 for

References

1. ^¹²³⁴⁵⁶⁷{{cite book|author=Huppert, B.|year=1979|title=Endliche Gruppen I|publisher=Grundlehren der mathematischen Wissenschaften, Vol. 134, Springer-Verlag Berlin Heidelberg New York}}
2. ^¹²³⁴{{cite journal|author=Schur, I.|year=1902| title=Neuer Beweis eines Satzes über endliche Gruppen|journal=Sitzungsb. Preuss. Akad. Wiss.|pages=1013–1019}}
3. ^¹²³{{cite journal|author=Artin, E.|year=1929| title=Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz|journal=Abh. Math. Sem. Univ. Hamburg|volume=7|pages=46–51}}
4. ^{{cite journal|author=Hasse, H.|year=1930| title=Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz|journal=Jahresber. Deutsch. Math. Verein., Ergänzungsband|volume=6|pages=1–204}}
5. ^¹²³{{cite book|author=Gorenstein, D.|year=2012|title=Finite groups|publisher=AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island}}
6. ^¹²³{{cite book|author=Hall M., jr.|year=1999|title=The theory of groups|publisher=AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island}}
7. ^¹²³{{cite book|author=Isaacs, I. M.|year=2008|title=Finite group theory|publisher=Graduate Studies in Mathematics, Vol. 92, American Mathematical Society, Providence, Rhode Island}}
8. ^¹²{{cite book|author=Aschbacher, M.|year=1986|title=Finite group theory|publisher=Cambridge Studies in Advanced Mathematics, Vol. 10, Cambridge University Press}}
9. ^¹²{{cite book|author=Smith, G. |author2=Tabachnikova, O.|year=2000|title=Topics in group theory|publisher=Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London}}
10. ^¹²{{cite journal|author=Blackburn, N.|year=1958|title=On a special class of p-groups|journal=Acta Math.|volume=100|pages=45–92|doi=10.1007/bf02559602}}
11. ^¹²³⁴⁵{{cite journal|author=Mayer, D. C.|year=2013| title=The distribution of second p-class groups on coclass graphs|journal=J. Théor. Nombres Bordeaux|volume=25|number=2|pages=401–456|doi=10.5802/jtnb.842|arxiv=1403.3833}}
12. ^{{cite journal|author=Chang, S. M. |author2=Foote, R.|year=1980| title=Capitulation in class field extensions of type (p,p)|journal=Can. J. Math.|volume=32|number=5|pages=1229–1243|doi=10.4153/cjm-1980-091-9}}
13. ^{{cite book|author=Besche, H. U. |author2=Eick, B. |author3=O'Brien, E. A.|year=2005|title=The SmallGroups Library – a library of groups of small order|publisher=An accepted and refereed GAP 4 package, available also in MAGMA}}
14. ^{{cite journal|author=Besche, H. U. |author2=Eick, B. |author3=O'Brien, E. A.|year=2002|title=A millennium project: constructing small groups|journal=Int. J. Algebra Comput.|volume=12|pages=623–644|doi=10.1142/s0218196702001115}}
15. ^¹²³{{cite journal|author=Bush, M. R. |author2=Mayer, D. C.|year=2015| title=3-class field towers of exact length 3|journal=J. Number Theory|volume=147|pages=766–777 (preprint: arXiv:1312.0251 [math.NT], 2013)|doi=10.1016/j.jnt.2014.08.010|arxiv=1312.0251}}
16. ^¹²³⁴{{cite journal|author=Mayer, D. C.|year=2012|title=Transfers of metabelian p-groups|journal=Monatsh. Math.|volume=166|number=3-4|pages=467–495|doi=10.1007/s00605-010-0277-x}}
17. ^¹²³⁴⁵⁶{{cite journal|author=Mayer, D. C.|year=2012|title=The second p-class group of a number field|journal=Int. J. Number Theory|volume=8|number=2|pages=471–505|doi=10.1142/s179304211250025x}}
18. ^{{cite journal|author=Scholz, A. |author2=Taussky, O.|year=1934| title=Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm|journal=J. Reine Angew. Math.|volume=171|pages=19–41}}
19. ^{{cite book|author=Newman, M. F.|year=1977|title=Determination of groups of prime-power order|publisher=pp. 73-84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin}}
20. ^{{cite journal|author=O'Brien, E. A.|year=1990|title=The p-group generation algorithm|journal=J. Symbolic Comput.|volume=9|pages=677–698|doi=10.1016/s0747-7171(08)80082-x}}
21. ^{{cite journal|author=Miech, R. J.|year=1970|title=Metabelian p-groups of maximal class|journal=Trans. Amer. Math. Soc.|volume=152|pages=331–373|doi=10.1090/s0002-9947-1970-0276343-7}}
22. ^¹²{{cite journal|author=Boston, N. |author2=Bush, M. R. |author3=Hajir, F.|year=2015|title=Heuristics for p-class towers of imaginary quadratic fields|journal=Math. Ann.|arxiv=1111.4679v2}}

Transversals of a subgroup

Permutation representation

Artin transfer

Independence of the transversal

Artin transfers as homomorphisms

Wreath product of H and S(n)

Composition of Artin transfers

Wreath product of S(m) and S(n)

Cycle decomposition

Transfer to a normal subgroup

Computational implementation

Abelianization of type (p,p)

Abelianization of type (p2,p)

Transfer kernels and targets

Abelianization of type (p,p)

Abelianization of type (p2,p)

First layer

Second layer

Transfer kernel type

Connections between layers

Inheritance from quotients

Passing through the abelianization

TTT singulets

TKT singulets

TTT and TKT multiplets

Inherited automorphisms

Stabilization criteria

Structured descendant trees (SDTs)

Pattern recognition

Historical example

Commutator calculus

Systematic library of SDTs

Coclass 1

Coclass 2

Abelianization of type (p,p)

Abelianization of type (p2,p)

Abelianization of type (p,p,p)

Coclass 3

Abelianization of type (p2,p)

Abelianization of type (p,p,p)

Arithmetical applications

Example

Comparison of various primes

References

Abelianization of type (p²,p)

Abelianization of type (p²,p)

Abelianization of type (p²,p)

Abelianization of type (p²,p)