Artin transfer (group theory)

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

is given by Finally, we want to emphasize the structural peculiarity of the monomial representation which corresponds to the composite of Artin transfers, defining for a permutation

Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product

th power Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels of Artin transfers from a group

, 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.

There are two options regarding the intermediate subgroups The common feature of all parent-descendant relations between finite p-groups is that the parent

[15] 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

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

Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws: A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applications to descendant trees.

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

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

The TTT of all groups in this tree represented by a big full disk, which indicates a bicyclic centre of type

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.

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

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 [18] tried to determine the Galois group

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

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

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

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.

For this purpose, we use the polynomial identity to obtain: Finally, we combine the results: generally and in particular, To determine the kernels, it remains to solve the equations: The following equivalences, for any

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

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

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

, which are known from Figures 8 and 9 already, and endow these trees with additional arithmetical structure by surrounding a realized vertex

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

of complex quadratic fields, it was proved[17] that only every other branch of the trees in Figures 10 and 11 can be populated by these metabelian

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

, this gives the relative frequency as an approximation to the asymptotic density of the population in Figure 12 and Table 3.

The dominant part of the second p-class groups of these fields populates the top vertices of order

(a finite union of descendant trees) with additional arithmetical structure by attaching the minimal absolute discriminant

FactorThroughAbelianization
Figure 1: Factoring through the abelianization.
EpiAndDerivedQuotients
Figure 2: Epimorphisms and derived quotients.
EpiAndArtinTransfers
Figure 3: Epimorphisms and Artin transfers.
TreeCoclass2RootQ
Figure 4: Endowing a descendant tree with information on Artin transfers.
Coclass1Tree2Groups
Figure 5: Structured descendant tree of 2-groups with coclass 1.
Coclass1Tree3Groups
Figure 6: Structured descendant tree of 3-groups with coclass 1.
Coclass1Tree5Groups
Figure 7: Structured descendant tree of metabelian 5-groups with coclass 1.
Coclass2TreeQType33
Figure 8: First structured descendant tree of 3-groups with coclass 2 and abelianization (3,3).
Coclass2TreeUType33
Figure 9: Second structured descendant tree of 3-groups with coclass 2 and abelianization (3,3).
MinDiscriminantsTreeQ
Figure 10: Minimal discriminants for the first ASCT of 3-groups with coclass 2 and abelianization (3,3).
MinDiscriminantsTreeU
Figure 11: Minimal discriminants for the second ASCT of 3-groups with coclass 2 and abelianization (3,3).
FrequencyCoclass2Type33Sporadic
Figure 12: Frequency of sporadic 3-groups with coclass 2 and abelianization (3,3).
MinDiscriminantsCoclass2Type33Sporadic
Figure 13: Minimal absolute discriminants of sporadic 3-groups with coclass 2 and abelianization (3,3).
MinDiscriminantsCoclass2Type55Sporadic
Figure 14: Minimal absolute discriminants of sporadic 5-groups with coclass 2 and abelianization (5,5).
MinDiscriminantsCoclass2Type77Sporadic
Figure 15: Minimal absolute discriminants of sporadic 7-groups with coclass 2 and abelianization (7,7).