A Counterexample to the First Zassenhaus Conjecture
Hans J. Zassenhaus conjectured that for any unit of finite order in the integral group ring of a finite group there exists a unit in the rational group algebra of such that for some . We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order whose integral group ring contains a unit of order which, in the rational group algebra, is not conjugate to any element of the form .
Let be a finite group and denote by the group ring of over a commutative ring . Denote by the unit group of . In the 1970’s Zassenhaus made three strong conjectures about finite subgroups of (cf. [Seh93, Section 37]). These conjectures, often called the first, second and third Zassenhaus conjecture and sometimes abbreviated as (ZC1), (ZC2) and (ZC3), had a lasting impact on research in the field. All three of these conjectures turned out to be true for nilpotent groups [Wei91], but metabelian counterexamples for the second and the third one were found by K. W. Roggenkamp and L. L. Scott [Sco92, Kli91]. Later M. Hertweck showed that there are counterexamples of order as small as [Her04, Section 11]. Unlike its siblings, the first Zassenhaus conjecture seemed to stand the test of time. Since it was the only one of the three to remain open, people in recent years started referring to it as the Zassenhaus conjecture, and we will do the same in this article.
If is a unit of finite order, then there is an such that
for some .
This conjecture first appeared in written form in [Zas74] and inspired a lot of research in the decades to follow. The first results on the conjecture were mostly concerned with special classes of metabelian groups, [HP72, PM73, AH80, LB83, RS83, PMS84, Mit86, PMRS86, SW86, MRSW87, LT90, LP92, LS98, JPM00, BHK04, RS06]. Almost all of these results were later generalised by Hertweck [Her06, Her08a]. Hertweck proved that the Zassenhaus conjecture holds for groups which have a normal Sylow -subgroup with abelian complement or a cyclic normal subgroup such that for some abelian subgroup of . The latter result was further generalised in [CMR13], proving that the Zassenhaus conjecture holds for cyclic-by-abelian groups. In a different vein, A. Weiss’ proof of the conjecture, or even a stronger version of it, for nilpotent groups [Wei88, Wei91], was certainly a highlight of the study. The conjecture is also known to hold for a few other classes of solvable groups [Fer87, DJ96, BKM16, MR17b, MR17c, MR17a], as well as for some small groups. In particular, the conjecture holds for groups of order smaller than [HK06, HS15, BHK17].
Progress on non-solvable groups was initially lagging. For many years the conjecture was only known to hold for the alternating and symmetric group of degree [LP89, LT91] and the special linear group [DJPM97]. This state of affairs changed when Hertweck introduced a method to tackle the conjecture involving Brauer characters [Her07]. Nevertheless, results are still relatively far and between [Her07, Her08b, BH08, Gil13, BM16, KK17, BM17], and, for instance, the only non-abelian simple groups for which the conjecture has been verified are the groups where , [BM16] or where is a Fermat or Mersenne prime [MRS16].
In the present article we show that the Zassenhaus conjecture is false by providing a series of metabelian groups such that contains a unit of finite order not conjugate in to any element of the form for .
Let us describe these groups. To this end, let and be odd primes, an odd divisor of and , the additive group , and let and be primitive elements in the multiplicative groups and , respectively. Consider the abelian group
There is an action of on given by
and we may form the semidirect product , which we denote by . The following are our main results:
Let , where is a root of the polynomial over and is a root of over . There exists a unit of order such that is not conjugate in to any element of the form for . In particular, the Zassenhaus conjecture does not hold for .
Let be an odd positive integer, and let be arbitrary. There exist infinitely many pairs of primes and such that, for any admissible choice of and , for there are , each of order , such that neither one of the is conjugate in to an element of the form for , or to any other for . In particular, the Zassenhaus conjecture does not hold for such a group .
A more precise version of Theorem B, specifying lower bounds for and as well as the rational conjugacy classes of the , can be found in Corollary 7.3. The idea that groups like might be good candidates for a counterexample to the Zassenhaus conjecture was noted in [MR17a]. Looking at the various positive results mentioned above, it seems that metabelian groups would have been the next logical step, and people working in the field certainly attempted to prove the Zassenhaus conjecture for metabelian groups, to no avail. What is more, the class of metabelian groups provided E. Dade’s counterexample to R. Brauer’s question, which asked whether for all fields implies that and are isomorphic [Dad71]. The second Zassenhaus conjecture mentioned above, which asked if different (normalised) group bases of are conjugate in , fails for metabelian groups as well [Kli91]. On the other hand, metabelian groups were one of the first classes of groups for which the isomorphism problem on integral group rings was known to have a positive answer [Whi68].
Here is an outline of our strategy to prove Theorems A and B:
If is a cyclic group of order , then a unit of order corresponds to a certain -module , called a “double action module”. This is the well-known double action formalism explained in Section 2, and the defining property of double action modules is that their restriction to is a free -module of rank one. This principle works for any commutative ring .
Once we fix a conjugacy class of units of order in , or equivalently a -double action module corresponding to it, we need to find a -lattice in whose restriction to is free.
Let denote the localisation of at the prime ideal . We provide a fairly general construction of double action modules over for groups of the form , where is abelian. This is done in Section 5, and, of course, subject to a whole list of conditions. The double action modules we construct are direct sums of direct summands of permutation modules (see Definition 5.4), and as a consequence the local version of the counterexample is fairly explicit (see Proposition 7.11 at the end).
The problem of turning a family of “compatible” -lattices in with free restriction to into a -lattice in with the same property can be solved using a rather general local-global principle, provided the centraliser of the unit is big enough (think of as already being fixed up to conjugacy in ). This is done in Section 6.
In the last section we study groups of the form as defined above. All of the more general results of the preceding sections become explicit and elementarily verifiable in this situation. We use the general result of that section, Theorem 7.2, to prove Theorems A and B.
In regard to future research, it seems worth pointing out that many variations and weaker versions of the Zassenhaus conjecture remain open. An overview of the weaker forms of the conjecture can be found in [MR17c]. In particular, the question if the orders of torsion units of augmentation one in coincide with the orders of elements in remains open. It also might still be true that if is a torsion unit then is conjugate in to for some , where is some larger group containing .
Going in a different direction, the -version of even the strongest of the three Zassenhaus conjectures remains open. This variation asks if it is true that any -subgroup of consisting of elements of augmentation one is conjugate in to a subgroup of . This is sometimes called “(p-ZC3)” or the “Strong Sylow Theorem” for . An overview of results relating to this problem can be found in [BKM16]. For the counterexample to the Zassenhaus conjecture presented in the present article it is of fundamental importance that the order of the unit is divisible by at least two different primes.
Throughout the paper we are going to use the following notation, most of which is quite standard.
Notation and basic definitions.
Let be a finite group and let be a subgroup of . For a character of we write for the induced character, and for a character of we write for the restriction to . The trivial character of is denoted by .
For a prime number we denote by a Sylow -subgroup of and by the -part of an element . The conjugacy class of is denoted by . We also use to denote a -Hall subgroup of (this is used only for nilpotent ) and for the -part of .
Let be a finite group and let and be subgroups of such that . If and are characters of and , respectively, then denotes the corresponding character of . Similarly, if and are -modules for some commutative ring , denotes with the natural -module structure.
We write “” to denote a sum ranging over a set of representatives of the conjugacy classes of . If acts on a set we write “” for the sum ranging over representatives of the -orbits in .
If and are groups, and acts on by automorphisms, we denote by the set of -orbits of irreducible rational characters of . We write “” for a sum ranging over representatives of these orbits.
For a cyclic group and an element we define
We will often use the fact that and for all primes .
If is a ring and is an -module we write for the direct sum of copies of .
If is a ring, is an -module and is an arbitrary subset, we write for the -module generated by .
Let be a commutative ring and let be an arbitrary element of . Then
is called the partial augmentation of at .
2. Double action formalism
The “double action formalism” (see, for instance, [Seh93, Section 38.6]) is a commonly used way of studying the Zassenhaus conjecture and other questions relating to units in group algebras via certain bimodules, the so-called “double action modules”. In this section we give a short (but complete, at least for our purposes) overview of this formalism. For the rest of this section let be a finite group and let be a cyclic group of order . By we denote an arbitrary commutative ring.
Given a unit satisfying we define an -module as follows: as an -module, is equal to , and the (right) action of is given by
where the product on the right hand side of the assignment is taken within the ring , and denotes the standard involution on . We call this -module the double action module associated with the unit .
An -module is called -regular if is free of rank one as an -module (that is, it is isomorphic to considered as a right module over itself).
A double action module is clearly -regular, but it turns out that the converse is true as well:
If is a -regular -module, then for some unit with .
If are two units satisfying , then
if and only if and are conjugate inside .
Assume that is -regular. Then we may choose an isomorphism of -modules , where we view as a right module over itself. As before, let denote the standard involution of the group algebra, and define . Then we have, for all ,
where we made use of the fact that . It now follows immediately from the above that , and the map is easily seen to be an isomorphism of -modules.
Let us now prove the second part of the proposition. To this end, fix an isomorphism . Then . As an equation purely in the ring this yields . So it only remains to show that is an invertible element of , which follows from the fact that generates as an -module. ∎
As we have seen so far, double action modules are in one-to-one correspondence with conjugacy classes of elements of whose order divides . Evidently this means that each property of torsion units should have a counterpart in the language of double action modules. An important tool in the study of the Zassenhaus conjecture is the criterion given in [MRSW87, Theorem 2.5]. It states that a unit of finite order and augmentation one is conjugate in to an element of if and only if for all and all . In particular, finding a counterexample to the Zassenhaus conjecture is equivalent to finding a unit of finite order and augmentation one which has a negative partial augmentation. Hence it is important for us to have a way of recovering the partial augmentations of a unit from the corresponding double action module.
Let be a unit satisfying . Let
denote the character of the -module . Then
Let us first calculate the trace of the linear map for arbitrary . This trace is equal to the number of such that , or, equivalently, . If then this number is equal to , otherwise it is zero. Now, if
then the linear endomorphism of induced by is equal to . The character value is the trace of this map, which is equal to
as claimed. ∎
We now turn our attention to the case of rational coefficients, i.e. . In that situation -regularity of -modules can readily be checked on the level of characters, and Proposition 2.3 can be used to ascertain whether the corresponding torsion unit in is indeed not conjugate to an element of .
Let be pair-wise non-conjugate elements of whose order divides , and let such that . Assume that
is in fact a character of , rather than just a virtual character. Then is the character of for some satisfying . Moreover for all and whenever is not conjugate to any of the .
Let us first prove that can be realised as the character of a -module, rather than just a -module. By definition can be written as the difference of the characters of two -modules, say and . Without loss of generality we may assume that and share no isomorphic simple direct summands. But then , which implies . That is, and share no isomorphic simple direct summands, which means that can only be a proper character if , which means that is a -module affording .
To verify that is the character of for some satisfying , it suffices to show that is equal to the regular character of , which is equal to . Note that by Mackey’s theorem we have
where ranges over a transversal for the double cosets . Since , there is just one such double coset, and therefore
independent of . Combining this fact with (5) we get
The character evaluated on is equal to the number of such that . Since the order of divides , which is the order of , it follows that the projection from to is an isomorphism, and hence is the only element of whose projection to is . This implies that if and only if . It follows that is equal to zero if , and equal to if . Plugging this back into (6) yields the desired result for the partial augmentations of . ∎
3. Local and semi-local rings of coefficients
Let be the ring of integers in an algebraic number field . For a maximal ideal of we let denote the localisation of at the prime . If is a finite collection of maximal ideals of , we define
which is a semi-local ring whose maximal ideals are precisely for . Testing whether a particular module is a double action module of a unit is particularly easy over , as the following proposition shows:
Let be an -module such that is projective and is -regular. Then is -regular.
By assumption is projective and its character is equal to the character of the regular -module. We need to show that this implies that is isomorphic to the regular -module. This follows from the fact that two projective -modules are isomorphic if and only if their characters are the same, a consequence of the fact that the decomposition matrix of a finite group has full row rank (see [CR81, Corollary 18.16] for the precise statement we are using). ∎
Constructing a -regular -module is actually equivalent to constructing a -regular -module for each in such a way that all of these modules have the same character.
Let be an -order in a finite-dimensional semisimple -algebra and let be a finite-dimensional -module.
Assume that we are given full -lattices for each . Then
is an -lattice in with the property that for each .
Given two -lattices and in , we have if and only if for each .
is clearly a -module, and in order to show that it is a lattice it suffices to show that it is contained in some -lattice. If is an arbitrary full -lattice in , then for each there is a number such that . Let be a number such that for each . Then for each , and therefore , which is an -lattice.
Now let us prove that for each . Clearly . On the other hand, if , then there is an integer such that , for all with , and (we can take to be contained in a product of sufficiently large powers of the maximal ideals in different from ). By definition, we now have , and since is invertible in we also have . This implies , which completes the proof of the first point. For the second point see [Rei75, Exercise 18.3]. ∎
4. Locally free lattices and class groups
As in the previous section let be the ring of integers in an algebraic number field . Let be a finite-dimensional semisimple -algebra, and let be an -order in . Throughout this section we adopt the following notational convention: if is a maximal ideal of , and is an -module, then denotes the -adic completion of . In particular, is a complete field with valuation ring , is a finite-dimensional -algebra and is an -order in .
Let us first check that no information is lost in passing from the localisations considered in the previous section to the completions we are going to consider now. If we keep the notation for the localisation of at and , then and can also be viewed as the -adic completions of and , respectively.
Proposition 4.1 ([Cr87, Proposition 30.17]).
Let and be finitely generated -modules. Then
In particular, if and are finitely generated -modules, then as -modules if and only if as -modules.
Now let us define the protagonist of this section: the locally free class group of .
Definition 4.2 (cf. [Cr87, §49a]).
A right -lattice is called locally free of rank if
as right -modules for all maximal ideals of .
If and are right -lattices, we say that and are stably isomorphic if
for some .
The locally free class group of , denoted by , is an additive group whose elements are the stable isomorphism classes of locally free right -ideals in . The group operation on is defined as follows: if and are locally free right -ideals in , then there is a locally free right -ideal such that
as right -modules. We define the sum to be equal to .
Note that the unit element of is . For the purposes of this article, class groups serve as a means to prove that certain -lattices are free. The reason this works is that most group algebras satisfy the Eichler condition relative to , which guarantees that we can infer from :
Definition 4.3 (cf. [Cr87, Remark 45.5 (i)]).
We say that satisfies the Eichler condition relative to if no simple component of is isomorphic to a totally definite quaternion algebra.
Theorem 4.4 (Jacobinski Cancellation Theorem [Cr87, Theorem 51.24]).
If satisfies the Eichler condition relative to , then any two locally free -lattices which are stably isomorphic are isomorphic.
Theorem 4.5 ([Cr87, Theorem 51.3]).
If is a finite group which does not have an epimorphic image isomorphic to either one of the following:
A generalised quaternion group of order where .
The binary tetrahedral group of order .
The binary octahedral group of order .
The binary icosahedral group of order .
then satisfies the Eichler condition relative to .
We now turn our attention to the problem of deciding whether a given locally free -ideal is trivial in .
Definition 4.6 ([Cr87, (49.4)]).
We define the idèle group of as
where ranges over all maximal ideals of . If and are two elements of , then their product in is defined as .
We identify with the subgroup of consisting of constant idèles.
This is also a subgroup of .
Even though it is not immediately obvious from the definition, does not depend on the order (in fact, if is another -order in , then for all except finitely many ).
Theorem 4.7 (Special case of [Cr87, Theorem 31.18]).
There is a bijection between the double cosets
and isomorphism classes of locally free right -ideals in given by
where . We denote the right hand side of this assignment by .
As shown in [CR87, Theorem 31.19] we have, for arbitrary , an isomorphism . This shows that there is an epimorphism of groups
Since is commutative by definition, we certainly have . In [Frö75], A. Fröhlich gave an explicit characterisation of the kernel of , which will be very useful to us later.
Definition 4.8 (Reduced norms).
Let be a field, and let be a finite-dimensional semisimple -algebra. Then there is a decomposition
where each is a simple -algebra. We may view as a central simple algebra over its centre . In each component we have a reduced norm map
obtained by embedding into for some field extension which splits , followed by mapping isomorphically onto a full matrix ring over of the appropriate dimension and then taking the determinant. We can then define a reduced norm map on component-wise. This map will take values in . That is, we get a multiplicative map
This is a normal subgroup of .
Theorem 4.10 ([Frö75, Theorem 1 and subsequent remarks]).
is an isomorphism of groups.
For our purposes it will suffice to know that for any the corresponding element is trivial.
5. Semi-local counterexamples
After these general sections we will now start to work with a more concrete class of groups which will ultimately provide our counterexample. Let be a finite group of the form
where is an abelian group. Moreover, let be a cyclic group such that the exponent of and the exponent of coincide. Let
be a class function which vanishes outside of (the notation for is deliberately chosen to resemble our notation for partial augmentations). When we say that the partial augmentations of a unit are given by , for some commutative ring , we mean that for all . Define
Note that, a priori, is only a virtual character of . Assume that all of the following hold:
If for some , then for all and all primes dividing the order of .
For each prime dividing the order of we have a decomposition
where is a proper character of for each .
The aim of this section is to prove the following theorem:
Let be a finite collection of primes. Then, under the above assumptions, there exists a -regular -lattice with character . Moreover, the partial augmentations of the associated unit are given by .
By [CW00, Theorem 3.3] the condition 3 actually implies that there exists a -lattice which is locally free over . We will not use this fact, but it provided the original motivation for this construction. The condition is also studied in [MR17a].
In (9), the are uniquely determined. Namely,
By Mackey decomposition we have
for every . Thus we obtain
Note that our assumptions imply that is equal to whenever . Hence
for every certainly ensures that (9) holds.
To prove that the ’s are uniquely determined as virtual characters, it suffices to show that they can be recovered from . Let . Then
Hence, if and , then
which shows that is determined uniquely by . ∎
For a group we define
Assume that is a prime dividing the order of and let be any prime not dividing the order of (in particular, is a possible choice for ). Let be a subgroup such that is cyclic. Let denote the primitive idempotent in the rational group algebra corresponding to the unique faithful irreducible representation of over , and denote its preimage in by (we may choose in such a way that ).
We define a -lattice
as well as a -lattice
The character of the -lattice is equal to
where is the unique irreducible rational character of with kernel .
This follows immediately from the fact that is afforded by the -lattice . ∎
The following description of the character of for certain is useful for explicit computations, even though we do not use it in this article:
If , then
If is cyclic of order for some prime , then
If , then is projective.
This follows from the Mackey formula, as is a direct summand of :
where the summation index runs over a transversal of the double cosets . Each summand on the right hand side is induced from a -subgroup of , and therefore is projective. ∎
Let be a prime dividing the order of and let be some -element of . Let be the following character of :
where is the character of defined in the beginning of this section (in particular is stabilised by ).
Then, for any prime not dividing the order of , is the restriction to of the character of the -lattice