Artin transfer (group theory)

In the mathematical field of group theory, an Artin transfer is a certain homomorphism 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 of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, 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.
Let formula_1 be a group and formula_2 be a subgroup of finite index formula_3
Definitions. A left transversal of formula_4 in formula_1 is an ordered system formula_6 of representatives for the left cosets of formula_4 in formula_1 such that
Similarly a right transversal of formula_4 in formula_1 is an ordered system formula_12 of representatives for the right cosets of formula_4 in formula_1 such that
Remark. For any transversal of formula_4 in formula_1, there exists a unique subscript formula_18 such that formula_19, resp. formula_20. Of course, this element with subscript formula_21 which represents the principal coset (i.e., the subgroup formula_4 itself) may be, but need not be, replaced by the neutral element formula_23.
Lemma. Let formula_1 be a non-abelian group with subgroup formula_4. Then the inverse elements formula_26 of a left transversal formula_6 of formula_4 in formula_1 form a right transversal of formula_4 in formula_1. Moreover, if formula_4 is a normal subgroup of formula_1, then any left transversal is also a right transversal of formula_4 in formula_1.
We must check when the image of a transversal under a homomorphism is also a transversal.
Proposition. Let formula_42 be a group homomorphism and formula_6 be a left transversal of a subgroup formula_4 in formula_1 with finite index formula_3 The following two conditions are equivalent:
Remark. We emphasize the important equivalence of the proposition in a formula:
Permutation representation.
Suppose formula_6 is a left transversal of a subgroup formula_4 of finite index formula_69 in a group formula_1. A fixed element formula_41 gives rise to a unique permutation formula_72 of the left cosets of formula_4 in formula_1 by left multiplication such that:
Using this we define a set of elements called the monomials associated with formula_76 with respect to formula_6:
Similarly, if formula_79 is a right transversal of formula_4 in formula_1, then a fixed element formula_41 gives rise to a unique permutation formula_83 of the right cosets of formula_4 in formula_1 by right multiplication such that:
And we define the monomials associated with formula_76 with respect to formula_79:
Definition. The mappings:
are called the permutation representation of formula_1 in the symmetric group formula_92 with respect to formula_6 and formula_79 respectively.
Definition. The mappings:
are called the monomial representation of formula_1 in formula_97 with respect to formula_6 and formula_79 respectively.
Lemma. For the right transversal formula_26 associated to the left transversal formula_101, we have the following relations between the monomials and permutations corresponding to an element formula_41:
Artin transfer.
Definitions. Let formula_1 be a group and formula_4 a subgroup of finite index formula_3 Assume formula_116 is a left transversal of formula_4 in formula_1 with associated permutation representation formula_119 such that
Similarly let formula_121 be a right transversal of formula_4 in formula_1 with associated permutation representation formula_124 such that
The Artin transfer formula_126 with respect to formula_6 is defined as:
Similarly we define:
Remarks. Isaacs calls the mappings
the "pre-transfer" from formula_1 to formula_4. The pre-transfer can be composed with a homomorphism formula_133 from formula_4 into an abelian group formula_135 to define a more "general version of the transfer" from formula_1 to formula_135 via formula_52, which occurs in the book by Gorenstein.
Taking the natural epimorphism
yields the preceding definition of the "Artin transfer" formula_141 in its original form by Schur and by Emil Artin, which has also been dubbed "Verlagerung" by Hasse. Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.
Independence of the transversal.
Proposition. The Artin transfers with respect to any two left transversals of formula_4 in formula_1 coincide.
Proposition. The Artin transfers with respect to any two right transversals of formula_4 in formula_1 coincide.
Proposition. The Artin transfers with respect to formula_166 and formula_116 coincide.
Corollary. The Artin transfer is independent of the choice of transversals and only depends on formula_4 and formula_1.
Artin transfers as homomorphisms.
Theorem. Let formula_6 be a left transversal of formula_4 in formula_1. The Artin transfer
and the permutation representation:
are group homomorphisms:
Let formula_178:
Since formula_160 is abelian and formula_181 is a permutation, we can change the order of the factors in the product:
This relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms.
It is illuminating to restate the homomorphism property of the Artin transfer in terms of the "monomial representation". The images of the factors formula_183 are given by
In the last proof, the image of the product formula_185 turned out to be
which is a very peculiar law of composition discussed in more detail in the following section.
The law is reminiscent of crossed homomorphisms formula_187 in the first cohomology group formula_188 of a formula_1-module formula_190, which have the property formula_191 for formula_178.
Wreath product of "H" and "S"("n").
The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian product formula_97 with a special law of composition known as the wreath product formula_194 of the groups formula_4 and formula_92 with respect to the set formula_197
Definition. For formula_178, the wreath product of the associated monomials and permutations is given by
Theorem. With this law of composition on formula_97 the "monomial representation" 
is an injective homomorphism.
The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group formula_97 endowed with the wreath product is given by formula_203, where the last formula_23 means the identity permutation. If formula_205, for some formula_41, then formula_207 and consequently
Finally, an application of the inverse inner automorphism with formula_209 yields formula_210, as required for injectivity.
Remark. The monomial representation of the theorem stands in contrast to the permutation representation, which cannot be injective if formula_211
Remark. Whereas Huppert uses the monomial representation for defining the Artin transfer, we prefer to give the immediate definitions in formulas (5) and (6) and to merely "illustrate" the homomorphism property of the Artin transfer with the aid of the monomial representation.
Composition of Artin transfers.
Theorem. Let formula_1 be a group with nested subgroups formula_213 such that formula_214 and formula_215 Then the Artin transfer formula_216 is the compositum of the induced transfer formula_217 and the Artin transfer formula_141, that is:
If formula_6 is a left transversal of formula_4 in formula_1 and formula_223 is a left transversal of formula_224 in formula_4, that is formula_226 and formula_227, then
is a disjoint left coset decomposition of formula_1 with respect to formula_224.
Given two elements formula_41 and formula_232, there exist unique permutations formula_72, and formula_234, such that
Then, anticipating the definition of the induced transfer, we have
For each pair of subscripts formula_106 and formula_238, we put formula_239, and we obtain
resp.
Therefore, the image of formula_76 under the Artin transfer formula_216 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 formula_247, and using the symbolic notation formula_248 for all pairs of subscripts formula_106, formula_238.
The preceding proof has shown that
Therefore, the action of the permutation formula_252 on the set formula_253 is given by formula_254. The action on the second component formula_255 depends on the first component formula_256 (via the permutation formula_257), whereas the action on the first component formula_256 is independent of the second component formula_255. Therefore, the permutation formula_247 can be identified with the multiplet
which will be written in twisted form in the next section.
Wreath product of "S"("m") and "S"("n").
The permutations formula_252, which arose as second components of the "monomial representation"
in the previous section, are of a very special kind. They belong to the "stabilizer" of the natural equipartition of the set formula_253 into the formula_69 rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this "stabilizer" is isomorphic to the wreath product formula_266 of the symmetric groups formula_267 and formula_92 with respect to the set formula_269, whose underlying set formula_270 is endowed with the following "law of composition":
This law reminds of the "chain rule" formula_272 for the "Fréchet derivative" in formula_273 of the compositum of "differentiable" functions formula_274 and formula_275 between "complete normed spaces".
The above considerations establish a third representation, the "stabilizer representation",
of the group formula_1 in the "wreath product" formula_266, similar to the "permutation representation" and the "monomial representation". As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if formula_1 is infinite. Formula (10) proves the following statement.
Theorem. The stabilizer representation
of the group formula_1 in the wreath product formula_266 of symmetric groups is a group homomorphism.
Cycle decomposition.
Let formula_6 be a left transversal of a subgroup formula_4 of finite index formula_69 in a group formula_1 and formula_287 be its associated permutation representation.
Theorem. Suppose the permutation formula_288 decomposes into pairwise disjoint (and thus commuting) cycles formula_289 of lengths formula_290 which is unique up to the ordering of the cycles. More explicitly, suppose
for formula_292, and formula_293 Then the image of formula_41 under the Artin transfer is given by
Define formula_296 for formula_297 and formula_292. This is a left transversal of formula_4 in formula_1 since
is a disjoint decomposition of formula_1 into left cosets of formula_4.
Fix a value of formula_292. Then:
Define:
Consequently,
The cycle decomposition corresponds to a formula_308 double coset decomposition of formula_1:
It was this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.
Transfer to a normal subgroup.
Let formula_4 be a normal subgroup of finite index formula_69 in a group formula_1. Then we have formula_314, for all formula_41, and there exists the quotient group formula_316 of order formula_69. For an element formula_318, we let formula_319 denote the order of the coset formula_320 in formula_316, and we let formula_322 be a left transversal of the subgroup formula_323 in formula_1, where formula_325.
Theorem. Then the image of formula_41 under the Artin transfer formula_141 is given by:
formula_329 is a cyclic subgroup of order formula_330 in formula_316, and a left transversal formula_322 of the subgroup formula_323 in formula_1, where formula_325 and formula_336 is the corresponding disjoint left coset decomposition, can be refined to a left transversal formula_337 with disjoint left coset decomposition:
of formula_4 in formula_1. Hence, the formula for the image of formula_76 under the Artin transfer formula_141 in the previous section takes the particular shape
with exponent formula_330 independent of formula_255.
Corollary. In particular, the inner transfer of an element formula_346 is given as a symbolic power:
with the trace element
of formula_4 in formula_1 as symbolic exponent.
The other extreme is the outer transfer of an element formula_61 which generates formula_316, that is formula_353.
It is simply an formula_69th power
The inner transfer of an element formula_346, whose coset formula_357 is the principal set in formula_316 of order formula_359, is given as the symbolic power
with the trace element
of formula_4 in formula_1 as symbolic exponent.
The outer transfer of an element formula_61 which generates formula_316, that is formula_366, whence the coset formula_320 is generator of formula_316 with orderformula_369, is given as the formula_69th 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 formula_1 to intermediate groups formula_373 between formula_1 and formula_375. For these intermediate groups we have the following lemma.
Lemma. All subgroups containing the commutator subgroup are normal.
Let formula_373. If formula_4 were not a normal subgroup of formula_1, then we had formula_379 for some element formula_61. This would imply the existence of elements formula_381 and formula_382 such that formula_383, and consequently the commutator formula_384 would be an element in formula_385 in contradiction to formula_386.
Explicit implementations of Artin transfers in the simplest situations are presented in the following section.
Computational implementation.
Abelianization of type ("p","p").
Let formula_1 be a "p"-group with abelianization formula_388 of elementary abelian type formula_389. Then formula_1 has formula_391 maximal subgroups formula_392 of index formula_393
Lemma. In this particular case, the Frattini subgroup, which is defined as the intersection of all maximal subgroups coincides with the commutator subgroup.
Proof. To see this note that due to the abelian type of formula_388 the commutator subgroup contains all "p"-th powers formula_395 and thus we have formula_396.
For each formula_397, let formula_398 be the Artin transfer homomorphism. According to Burnside's basis theorem the group formula_1 can therefore be generated by two elements formula_183 such that formula_401 For each of the maximal subgroups formula_402, which are also normal we need a generator formula_403 with respect to formula_375, and a generator formula_405 of a transversal formula_406 such that
A convenient selection is given by
Then, for each formula_397 we use equations (16) and (18) to implement the inner and outer transfers:
The reason is that in formula_411 formula_412 and formula_413
The complete specification of the Artin transfers formula_414 also requires explicit knowledge of the derived subgroups formula_415. Since formula_375 is a normal subgroup of index formula_417 in formula_402, a certain general reduction is possible by formula_419 but a presentation of formula_1 must be known for determining generators of formula_421, whence
Abelianization of type ("p"2,"p").
Let formula_1 be a "p"-group with abelianization formula_388 of non-elementary abelian type formula_425. Then formula_1 has formula_391 maximal subgroups formula_392 of index formula_417 and formula_391 subgroups formula_431 of index formula_432 For each formula_433 let
be the Artin transfer homomorphisms. Burnside's basis theorem asserts that the group formula_1 can be generated by two elements formula_183 such that formula_437
We begin by considering the first layer of subgroups. For each of the normal subgroups formula_402, we select a generator
such that formula_440. These are the cases where the factor group formula_441 is cyclic of order formula_442. However, for the distinguished maximal subgroup formula_443, for which the factor group formula_444 is bicyclic of type formula_389, we need two generators:
such that formula_447. Further, a generator formula_405 of a transversal must be given such that formula_449, for each formula_397. It is convenient to define
Then, for each formula_397, we have inner and outer transfers:
since formula_412 and formula_455.
Now we continue by considering the second layer of subgroups. For each of the normal subgroups formula_456, we select a generator
such that formula_458. Among these subgroups, the Frattini subgroup formula_459 is particularly distinguished. A uniform way of defining generators formula_460 of a transversal such that formula_461, is to set
Since formula_463, but on the other hand formula_464 and formula_465, for formula_397, with the single exception that formula_467, we obtain the following expressions for the inner and outer transfers
exceptionally
The structure of the derived subgroups formula_415 and formula_471 must be known to specify the action of the Artin transfers completely.
Transfer kernels and targets.
Let formula_1 be a group with finite abelianization formula_388. Suppose that formula_474 denotes the family of all subgroups which contain formula_375 and are therefore necessarily normal, enumerated by a finite index set formula_476. For each formula_477, let formula_478 be the Artin transfer from formula_1 to the abelianization formula_480.
Definition. The family of normal subgroups formula_481 is called the transfer kernel type (TKT) of formula_1 with respect to formula_474, and the family of abelianizations (resp. their abelian type invariants) formula_484 is called the transfer target type (TTT) of formula_1 with respect to formula_474. 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 formula_1 be a "p"-group with abelianization formula_388 of elementary abelian type formula_389. Then formula_1 has formula_391 maximal subgroups formula_392 of index formula_417. For formula_433 let formula_398 denote the Artin transfer homomorphism.
Definition. The family of normal subgroups formula_496 is called the transfer kernel type (TKT) of formula_1 with respect to formula_498.
Remark. For brevity, the TKT is identified with the multiplet formula_499, whose integer components are given by
Here, we take into consideration that each transfer kernel formula_501 must contain the commutator subgroup formula_375 of formula_1, since the transfer target formula_480 is abelian. However, the minimal case formula_505 cannot occur.
Remark. A renumeration of the maximal subgroups formula_506 and of the transfers formula_507 by means of a permutation formula_508 gives rise to a new TKT formula_509 with respect to formula_510, identified with formula_511, where
It is adequate to view the TKTs formula_513 as equivalent. Since we have
the relation between formula_515 and formula_516 is given by formula_517. Therefore, formula_515 is another representative of the orbit formula_519 of formula_516 under the action formula_521 of the symmetric group formula_522 on the set of all mappings from formula_523 where the extension formula_524 of the permutation formula_508 is defined by formula_526 and formally formula_527
Definition. The orbit formula_528 of any representative formula_516 is an invariant of the "p"-group formula_1 and is called its transfer kernel type, briefly TKT.
Remark. Let formula_531 denote the counter of total transfer kernels formula_532, which is an invariant of the group formula_1. In 1980, S. M. Chang and R. Foote proved that, for any odd prime formula_417 and for any integer formula_535, there exist metabelian "p"-groups formula_1 having abelianization formula_388 of type formula_389 such that formula_539. However, for formula_540, there do not exist non-abelian formula_541-groups formula_1 with formula_543, which must be metabelian of maximal class, such that formula_544. Only the elementary abelian formula_541-group formula_546 has formula_547. See Figure 5.
In the following concrete examples for the counters formula_548, 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.
For formula_549, we have
Abelianization of type ("p"2,"p").
Let formula_1 be a "p"-group with abelianization formula_388 of non-elementary abelian type formula_569 Then formula_1 possesses formula_391 maximal subgroups formula_392 of index formula_417 and formula_391 subgroups formula_431 of index formula_432
Assumption. Suppose
is the distinguished maximal subgroup and
is the distinguished subgroup of index formula_442 which as the intersection of all maximal subgroups, is the Frattini subgroup formula_580 of formula_1.
First layer.
For each formula_397, let formula_583 denote the Artin transfer homomorphism.
Definition. The family formula_584 is called the first layer transfer kernel type of formula_1 with respect to formula_498 and formula_587, and is identified with formula_588, where
Remark. Here, we observe that each first layer transfer kernel is of exponent formula_417 with respect to formula_375 and consequently cannot coincide with formula_592 for any formula_593, since formula_594 is cyclic of order formula_442, whereas formula_444 is bicyclic of type formula_389.
Second layer.
For each formula_397, let formula_599 be the Artin transfer homomorphism from formula_1 to the abelianization of formula_456.
Definition. The family formula_602 is called the second layer transfer kernel type of formula_1 with respect to formula_587 and formula_498, and is identified with formula_606 where
Transfer kernel type.
Combining the information on the two layers, we obtain the (complete) transfer kernel type formula_608 of the "p"-group formula_1 with respect to formula_498 and formula_587.
Remark. The distinguished subgroups formula_443 and formula_613 are unique invariants of formula_1 and should not be renumerated. However, independent renumerations of the remaining maximal subgroups formula_615 and the transfers formula_616 by means of a permutation formula_617, and of the remaining subgroups formula_618 of index formula_442 and the transfers formula_620 by means of a permutation formula_621, give rise to new TKTs formula_622 with respect to formula_510 and formula_624, identified with formula_625, where
and formula_627 with respect to formula_628 and formula_510, identified with formula_630 where
It is adequate to view the TKTs formula_632 and formula_633 as equivalent. Since we have
the relations between formula_635 and formula_636, and formula_637 and formula_638, are given by
Therefore, formula_641 is another representative of the orbit formula_642 of formula_643 under the action:
of the product of two symmetric groups formula_645 on the set of all pairs of mappings formula_646, where the extensions formula_647 and formula_524 of a permutation formula_649 are defined by formula_650 and formula_651, and formally formula_652 and formula_653
Definition. The orbit formula_654 of any representative formula_655 is an invariant of the "p"-group formula_1 and is called its transfer kernel type, briefly TKT.
Connections between layers.
The Artin transfer formula_599 is the composition formula_658 of the induced transfer formula_659 from formula_592 to formula_456 and the Artin transfer formula_662
There are two options regarding the intermediate subgroups
Inheritance from quotients.
The common feature of all parent-descendant relations between finite "p"-groups is that the parent formula_688 is a quotient formula_689 of the descendant formula_1 by a suitable normal subgroup formula_691 Thus, an equivalent definition can be given by selecting an epimorphism formula_692 with formula_693 Then the group formula_694 can be viewed as the parent of the descendant formula_1.
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.
Proof. 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 formula_702 is a consequence of the condition formula_703 which implies for any formula_41 we have:
formula_702 is a homomorphism, let formula_178 be arbitrary, then:
Thus, the commutator subgroup formula_709, and this finally shows that the definition of formula_702 is independent of the coset representative,
TTT singulets.
Proof. This claim is a consequence of the Main Theorem in the article on the induced homomorphism. Nevertheless, an independent proof is given as follows: first, the image of the commutator subgroup is
Second, the epimorphism formula_52 can be restricted to an epimorphism formula_725. According to the previous section, the composite epimorphism formula_726 factors through formula_160 by means of a uniquely determined epimorphism formula_728 such that formula_729. Consequently, we have formula_730. Furthermore, the kernel of formula_702 is given explicitly by formula_732.
Finally, if formula_721, then formula_702 is an isomorphism, since formula_735.
Definition. 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 formula_736, when formula_730, and formula_738, when formula_739.
TKT singulets.
Proof. Let formula_6 be a left transversal of formula_4 in formula_1. Then we have a disjoint union:
Consider the image of this disjoint union, which is not necessarily disjoint,
and let formula_758 We have:
Let formula_728 be the epimorphism from the previous proposition. We have:
Since formula_762, the right hand side equals formula_763, if formula_47 is a left transversal of formula_715 in formula_749, which is true when formula_767 Therefore, formula_768 Consequently, formula_769 implies the inclusion
Finally, if formula_771, then by the previous proposition formula_702 is an isomorphism. Using its inverse we get formula_773, which proves
Combining the inclusions we have:
Definition. In view of the results in the present section, we are able to define a partial order of transfer kernels by setting formula_776, when formula_750
TTT and TKT multiplets.
Assume formula_712 are as above and that formula_388 and formula_780 are isomorphic and finite. Let formula_474 denote the family of all subgroups containing formula_375 (making it a finite family of normal subgroups). For each formula_477 let:
Take formula_785 be any non-empty subset of formula_476. Then it is convenient to define formula_787, called the (partial) transfer kernel type (TKT) of formula_1 with respect to formula_789, and formula_790 called the (partial) transfer target type (TTT) of formula_1 with respect to formula_789.
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.
Proof. Using the isomorphism formula_814 we define:
First we show this map is well-defined:
The fact that formula_817 is surjective, a homomorphism and satisfies formula_811 are easily verified.
And if formula_819, then injectivity of formula_817 is a consequence of
Let formula_698 be the canonical projection then there exists a unique induced automorphism formula_823 such that formula_824, that is,
The reason for the injectivity of formula_826 is that
since formula_375 is a characteristic subgroup of formula_1.
Definition. formula_1 is called a "σ"−group, if there exists formula_831 such that the induced automorphism acts like the inversion on formula_388, that is for all
The Inheritance Law III asserts that, if formula_1 is a "σ"−group and formula_835, then formula_749 is also a "σ"−group, the required automorphism being formula_817. This can be seen by applying the epimorphism formula_52 to the equation formula_839 which yields
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
Assumption. The parent formula_688 of a group formula_1 is the quotient formula_843 of formula_1 by the last non-trivial term formula_845 of the lower central series of formula_1, where formula_847 denotes the nilpotency class of formula_1. The corresponding epimorphism formula_849 from formula_1 onto formula_851 is the canonical projection, whose kernel is given by formula_852.
Under this assumption,
the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite "p"-groups.
Compatibility criterion. Let formula_417 be a prime number. Suppose that formula_1 is a non-abelian finite "p"-group of nilpotency class formula_855. Then the TTT and the TKT of formula_1 and of its parent formula_688 are comparable in the sense that formula_858 and formula_859.
The simple reason for this fact is that, for any subgroup formula_373, we have formula_861, since formula_862.
For the remaining part of this section, the investigated groups are supposed to be finite metabelian "p"-groups formula_1 with elementary abelianization formula_388 of rank formula_541, that is of type formula_389.
Partial stabilization for maximal class. A metabelian "p"-group formula_1 of coclass formula_868 and of nilpotency class formula_869 shares the last formula_417 components of the TTT formula_871 and of the TKT formula_872 with its parent formula_688. More explicitly, for odd primes formula_874, we have formula_875 and formula_876 for formula_877.
This criterion is due to the fact that formula_878 implies formula_879,
for the last formula_417 maximal subgroups formula_881 of formula_1.
The condition formula_878 is indeed necessary for the partial stabilization criterion. For odd primes formula_874, the extra special formula_417-group formula_886 of order formula_887 and exponent formula_442 has nilpotency class formula_889 only, and the last formula_417 components of its TKT formula_891 are strictly smaller than the corresponding components of the TKT formula_892 of its parent formula_688 which is the elementary abelian formula_417-group of type formula_389.
For formula_540, both extra special formula_541-groups of coclass formula_23 and class formula_889, the ordinary quaternion group formula_886 with TKT formula_901 and the dihedral group formula_902 with TKT formula_903, have strictly smaller last two components of their TKTs than their common parent formula_904 with TKT formula_905.
Total stabilization for maximal class and positive defect.
A metabelian "p"-group formula_1 of coclass formula_868 and of nilpotency class formula_908, that is, with index of nilpotency formula_909, shares all formula_391 components of the TTT formula_871 and of the TKT formula_872 with its parent formula_688, provided it has positive defect of commutativity formula_914.
Note that formula_915 implies formula_874, and we have formula_876 for all formula_397.
This statement can be seen by observing that the conditions formula_909 and formula_915 imply formula_921,
for all the formula_391 maximal subgroups formula_498 of formula_1.
The condition formula_915 is indeed necessary for total stabilization. To see this it suffices to consider the first component of the TKT only. For each nilpotency class formula_926, there exist (at least) two groups formula_927 with TKT formula_928 and formula_929 with TKT formula_930, both with defect formula_931, where the first component of their TKT is strictly smaller than the first component of the TKT formula_892 of their common parent formula_933.
Partial stabilization for non-maximal class.
Let formula_549 be fixed. A metabelian 3-group formula_1 with abelianization formula_936, coclass formula_937 and nilpotency class formula_938 shares the last two (among the four) components of the TTT formula_871 and of the TKT formula_872 with its parent formula_688.
This criterion is justified by the following consideration. If formula_926, then formula_943
for the last two maximal subgroups formula_944 of formula_1.
The condition formula_926 is indeed unavoidable for partial stabilization, since there exist several formula_565-groups of class formula_948, for instance those with SmallGroups identifiers formula_949, such that the last two components of their TKTs formula_950 are strictly smaller than the last two components of the TKT formula_566 of their common parent formula_952.
Total stabilization for non-maximal class and cyclic centre.
Again, let formula_549 be fixed.
A metabelian 3-group formula_1 with abelianization formula_936, coclass formula_937, nilpotency class formula_938 and cyclic centre formula_958 shares all four components of the TTT formula_871 and of the TKT formula_872 with its parent formula_688.
The reason is that, due to the cyclic centre, we have formula_962
for all four maximal subgroups formula_963 of formula_1.
The condition of a cyclic centre is indeed necessary for total stabilization, since for a group with bicyclic centre there occur two possibilities.
Either formula_965 is also bicyclic, whence formula_966 is never contained in formula_967,
or formula_968 is cyclic but is never contained in formula_969.
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 formula_549, and the chosen descendant tree is actually a coclass tree having a unique infinite mainline, branches of depth formula_565, and strict periodicity of length formula_541 setting in with branch formula_973.
The initial pre-period consists of branches formula_974 and formula_975 with exceptional structure.
Branches formula_973 and formula_977 form the primitive period such that formula_978, for odd formula_979, and formula_980, for even formula_981.
The root of the tree is the metabelian formula_565-group with identifier formula_983, that is, a group of order formula_984 and with counting number formula_985. This root is not coclass settled, whence its entire descendant tree formula_986 is of considerably higher complexity than the coclass-formula_541 subtree formula_988, 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) formula_516, and the vertical ordinate is labelled with a single component formula_990 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 formula_23, 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 formula_541, 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 formula_993, is given by formula_994 with varying first component formula_995, the nearly homocyclic abelian formula_565-group of order formula_997, and fixed further components formula_998 and formula_999, where the abelian type invariants are either written as orders of cyclic components or as their formula_565-logarithms with exponents indicating iteration. (The latter notation is employed in Figure 4.) Since the coclass of all groups in this tree is formula_541, the connection between the order formula_1002 and the nilpotency class is given by formula_1003.
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 formula_161-groups in Figure 4 nor to eliminate all groups with TKT formula_1006.
In Figure 4, the big double contour rectangle surrounds the pruned coclass tree formula_1007, where the numerous vertices with TKT formula_1008 are completely eliminated. This would, for instance, be useful for searching a formula_161-group with TKT formula_1010 and first component formula_1011 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 formula_1012 of the Hilbert formula_565-class field tower, that is the maximal unramified pro-formula_565 extension formula_1015, of the complex quadratic number field formula_1016 They actually succeeded in finding the maximal metabelian quotient formula_1017 of formula_1, that is the Galois group of the second Hilbert formula_565-class field formula_1020 of formula_224.
However, it needed formula_1022 years until M. R. Bush and D. C. Mayer, in 2012, provided the first rigorous proof
that the (potentially infinite) formula_565-tower group formula_1024 coincides with the finite formula_565-group formula_1026 of derived length formula_1027, and thus the formula_565-tower of formula_224 has exactly three stages, stopping at the third Hilbert formula_565-class field formula_1031 of formula_224.
In contrast, let us secondly select the sporadic part formula_1033 of the coclass graph formula_1034 for demonstrating that another way of attaching additional "arithmetical structure" to descendant trees is to display the counter formula_1035 by the second formula_565-class group formula_1037 of fields with absolute discriminants below a given upper bound formula_1038, for instance formula_1039. With respect to the total counter formula_1040 of all complex quadratic fields with formula_565-class group of type formula_993 and discriminant formula_1043, this gives the relative frequency as an approximation to the asymptotic density of the population in Figure 12 and Table 3. Exactly four vertices of the finite sporadic part formula_1033 of formula_1034 are populated by second formula_565-class groups formula_1037:
Comparison of various primes.
Now let formula_1052 and consider complex quadratic fields formula_1053 with fixed signature formula_1054 and "p"-class groups of type formula_389. The dominant part of the second "p"-class groups of these fields populates the top vertices of order formula_1056 of the sporadic part formula_1057 of the coclass graph formula_1058, which belong to the stem of P. Hall's isoclinism family formula_1059, or their immediate descendants of order formula_1060. For primes formula_1061, the stem of formula_1059 consists of formula_1063 regular "p"-groups and reveals a rather uniform behaviour with respect to TKTs and TTTs, but the seven formula_565-groups in the stem of formula_1059 are irregular. We emphasize that there also exist several (formula_565 for formula_549 and formula_1068 for formula_1061) infinitely capable vertices in the stem of formula_1059 which are partially roots of coclass trees. However, here we focus on the sporadic vertices which are either isolated Schur formula_161-groups (formula_541 for formula_549 and formula_391 for formula_1061) or roots of finite trees within formula_1057 (formula_541 for each formula_874). For formula_1061, the TKT of Schur formula_161-groups is a permutation whose cycle decomposition does not contain transpositions, whereas the TKT of roots of finite trees is a compositum of disjoint transpositions having an even number (formula_1081 or formula_541) of fixed points.
We endow the "forest" formula_1057 (a finite union of descendant trees) with additional "arithmetical structure" by attaching the "minimal absolute discriminant" formula_1084 to each "realized" vertex formula_1085. The resulting structured sporadic coclass graph is shown in Figure 13 for formula_549, in Figure 14 for formula_1087, and in Figure 15 for formula_1088.