Rational conjugacy of torsion units
in integral group rings
of non-solvable groups00footnotetext: Mathematics Subject Classification. 16u60, 16s34, 20c05, 20c10
Key words and phrases. Integral group ring, torsion unit, Zassenhaus Conjecture, Prime graph question.
Abstract. We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group . We also prove the Zassenhaus Conjecture for . In a second application we show that there are no normalized units of order in the integral group rings of and . This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative answer for all groups having an order divisible by at most three different primes.
Througout this paper let be a finite group, the integral group ring of and the group of augmentation one units in , aka. normalized units. The most famous open conjecture regarding torsion units in is:
The Zassenhaus Conjecture (ZC): Let be a torsion unit. Then there exist a unit and such that
If for a unit such and exist we say that is rationally conjugate to . There are positive results for the Zassenhaus Conjecture for classes of solvable groups (e.g. A. Weiss proved it for nilpotent groups [Wei91] and M. Caicedo, L. Margolis and Á. del Río established it for all cyclic-by-abelian groups [CMdR13]). For non-solvable groups it is only known for specific groups, e.g. for [LP89], [LT91], [Her08], or for a prime [Her07, KK13, Gil13].
The Prime Graph Question (PQ): Let and be different primes such that has a unit of order . Does this imply that has an element of that order?
This is the same as to ask whether and have the same prime graph. Much more is known here: e.g. it has an affirmative answer for all solvable groups [Kim06] or the series , a prime [Her07]. V. Bovdi and A. Konovalov with different collaborators obtained positive answers to (PQ) for many of the sporadic simple groups, see e.g. [BK12] for recent results. Lately substantial progress was made, when W. Kimmerle and A. Konovalov obtained the first reduction result for the Prime Graph Question [KK13, Proposition 4.1] (cf. also [KK12, Theorem 2.1]):
Theorem (Kimmerle, Konovalov).
If (PQ) has affirmative answer for all almost-simple images of , then it also has an affirmative answer for itself.
Recall that a group is almost simple if for a simple group . Using the above theorem, they proved that the Prime Graph Question has a positive answer for all finite groups whose order is divisible by at most three different primes, if it has a positive answer for and [KK12, Theorem 3.1]. Their result also places special emphasis on investigating the Prime Graph Question for almost simple groups.
All proofs of (ZC) for non-solvable groups rely on the so-called Luthar-Passi-Hertweck-method [LP89, Her07], referred to as the HeLP-method. But in many cases this method does not suffice to prove (ZC), e.g. it fails for [Her08], (see below) or [BK07]. Sometimes special arguments were considered in such situations as in [LT91], [Her06, Example 2.6], or [Her08]. But these arguments were designed for very special situations and are hard to generalize or seem not to give new information in other situations.
In this paper we introduce a new method to study rational conjugacy of torsion units inspired by M. Hertweck’s arguments for proving (ZC) for the alternating group of degree 6 [Her08]. This method is especially interesting for units of mixed order (i.e. not of prime power order) and in combination with the HeLP-method. We then give two applications of this method to prove:
The Zassenhaus Conjecture holds for the groups and
((ZC) for is proved using known methods.)
There are no units of order 6 in and in Here denotes the Mathieu group of degree 10.
Let be a finite group. Suppose that the order of all almost simple images of is divisible by at most three different primes. Then the prime graph of the normalized units of coincides with that of . In particular, the Prime Graph Question has a positive answer for all groups with an order divisible by at most three different primes.
1 From eigenvalues under ordinary representations to the modular module structure
Let be a finite group. The main tool to study rational conjugacy of torsion units are partial augmentations: Let and be the conjugacy class of the element in Then is called the partial augmentation of at This relates to (ZC) via:
Lemma 1.1 (Marciniak, Ritter, Sehgal, Weiss).
Let be a torsion unit of order . Then is rationally conjugate to a group element if and only if for all and all powers of with .
Proof. [MRSW87, Theorem 2.5].
It is well known that if is a torsion unit in , then by the so-called Berman-Higman Theorem [Seh93, Proposition 1.4]. If , then the order of divides the order of [MRSW87, Theorem 2.7], [Her06, Proposition 3.1]. Moreover the exponents of and of coincide [CL65]. We will use this in the following without further mention.
Let be a field, a -representation of with corresponding character and a torsion unit of order . If and all partial augmentations of and all its powers are known, and the characteristic of does not divide , we can compute the eigenvalues of in a field extension of which is large enough (a field which is a splitting field for and all its subgroups will do; there will be plenty of examples for this kind of calculations in §2). The HeLP-method makes use of the fact that the multiplicity of each -th root of unity as an eigenvalue of is a non-negative integer.
Notations: will always denote a prime, the -adic completion of and the ring of integers of . By we denote a complete local ring with maximal ideal containing and by the field of fractions of Denote by a finite field of characteritic containing the residue class field of . The reduction modulo , also with respect to lattices, will be denoted by .
The idea of our method is that if is an -representation of a group with corresponding -lattice and is a torsion unit in of order divisible by , we can reduce modulo and obtain restrictions on the isomorphism type of -modules considered as -modules, where is big enough to allow realizations of all irreducible -modular representations of Note that the Krull-Schmidt-Azumaya Theorem holds for finitely generated -lattices [CR81, Theorem 30.6]. From these isomorphism types we can then obtain restrictions on the isomorphism types of the -composition factors of when viewed as -module. Since a simple -module may appear in the reduction of several ordinary representations, this may finally yield a contradiction to the existence of A rough sketch of the method is given in Figure 1. As the direct path from the eigenvalues of an ordinary representation to the isomorphism types of the corresponding reduced module is not always evident, we are sometimes forced to take take the detour along the dashed arrows.
of *\frm<8pt>- \ar@->[r] \ar@–>[d] & *+\txtPossible isomorphism types
of as -module \ar[r] & \txtPossible isomorphism type of
-composition factors of
*+\txtComposition factors of
as -module \ar@–>[r] & *+\txtPossible isomorphism types
of as -lattice \ar@–>[u]
The connections between the eigenvalues of ordinary representations and the isomorphism type of the modules in positive characteristic for some cases are contained in the following propositions, which are consequences of known modular and integral representation theory.
The first proposition is standard knowledge in modular representation theory and may be found in e.g. [HB82, Theorem 5.3, Theorem 5.5].
Let be a cyclic group of order , where does not divide . Let be a field of characteristic containing a primitive -th root of unity . Then:
Up to isomorphism there are simple -modules. All these modules are one-dimensional as -vector spaces, acts trivially on them and acts as for . We denote these modules by .
The projective indecomposable -modules are of dimension . They are all uniserial and all composition factors of a projective indecomposable -module are isomorphic. There are non-isomorphic projective indecomposable -modules.
Each indecomposable -module is isomorphic to a submodule of a projective indecomposable module. So there are indecomposable modules, which are all uniserial and all composition factors of an indecomposable -module are isomorphic.
Let be a cyclic group of order , where does not divide . Let be a complete local ring containing a primitive -th root of unity . Let be an -representation of and let be an -lattice affording this representation.
Let be sets with multiplicities of -th roots of unity such that are the complex eigenvalues of where is possible. Let be -modules such that if is a representation of affording the eigenvalues of are Then
such that (The superscripts are merely meant as indices.) Moreover and the only composition factor of is , see the notation in Proposition 1.2.
To understand the full connection between the eigenvalues and the structure of , i.e. to follow the arrow in the second line of Figure 1, one must study the representation theory of The representation type of may be finite, tame or wild. Roughly speaking, the representation theory gets more complicated with increasing and increasing ramification index of over A listing of all representation types may be found in [Die85]. Some results concerning the connection between and are recorded in the next propositions. The first one is a consequence of [HR62, Theorem 2.6].
Let the notation be as in Proposition 1.3, assume and that is unramified over . Let be a primitive -th root of unity. Up to isomorphism there are 3 indecomposable -lattices . Each remains indecomposable. The -rank and the corresponding eigenvalues of are: with eigenvalue , with eigenvalues and with eigenvalues .
Notation: We denote the indecomposable lattices in Proposition 1.4 with their natural names: the trivial lattice , the augmentation ideal and the group ring .
When considering -lattices or -modules as in Proposition 1.3 (i.e. and ), we abbreviate the superscript and to and , respectively, i.e. , , and for the direct summands having trivial and non-trivial composition factors, respectively.
Let the notation be as in Proposition 1.3, be odd and with Assume is the -adic completion of an extension of which is unramified at . Denote by the ring of integers of and let . Note that there are exactly three simple -lattices up to isomorphism. The two following facts hold for indecomposable -lattices:
If is an indecomposable -lattice, then the non-trivial simple lattices each appear at most once as a composition factor of L, and the trivial one at most twice.
If is an indecomposable -lattice having at most two non-isomorphic composition factors, then each composition factor appears at most once.
If is the -adic completion of and is its ring of integers, then up to isomorphism all indecomposable -lattices are explicitly given in [Gud67, Lemma 4.1]. These lattices satisfy the statements of the proposition. If is an indecomposable -lattice, it is a direct summand of , where is an indecomposable -lattice, by the last paragraph of the proof of [CR81, Proposition 33.16]. So the statements of the proposition still hold for indecomposable -lattices. ∎
For a group we denote by an ordinary character of and by a representation of affording this character. By we denote a Brauer character and by a representation affording We write or to indicate that are the eigenvalues (with multiplicities) of the corresponding matrix. To improve readability, we sometimes group the eigenvalues appearing several times, e.g. indicates that has three times and and each twice as eigenvalue. By we will denote some fixed primitive complex -th root of unity. Especially we will use to denote the eigenvalues of a matrix of finite order over a field of characteristic where is prime to in the sense of Brauer.
Let be an algebraically closed field, a -representation of with character and a torsion unit in such that the characteristic of does not divide the order of . Let and be natural numbers such that . Let and As and are simultaneously diagonalizable over this means with On the other hand , where the sum runs over all conjugacy classes of . Comparing these two computations is the basic idea of the HeLP-method. We will use it freely in the following computations.
2.1 The groups . Proof of Theorem 1.
For the rest of the section let be a prime. Rational conjugacy of torsion units in integral group rings of the groups were studied by Hertweck in [Her07]. Combining some propositions from that note we directly obtain:
Proposition 2.1 (Hertweck).
Let and a torsion unit in .
If is of order prime to there exists an element in of the same order as If moreover the order of is prime, is rationally conjugate to an element in
If and divides the order of , then is of order and rationally conjugate to an element in
Assume is neither 2 nor 3 and is of order 6. Then is rationally conjugate to an element in .
Proof. [Her07, Propositions 6.1, 6.3, 6.4, 6.6 and 6.7].
The HeLP-method verifies the Zassenhaus Conjecture for if . We give a quick account: (ZC) is solved for already in [HP72], in [AH80], in [LP89], in [Her06], in [Her07] and independently in [KK13] and [Gil13]. The HeLP-method also suffices to prove (ZC) for (see below), but not for We will always use the character tables and Brauer tables from the ATLAS [WWT]111All tables used are accessible in GAP [GAP14] via the commands CharacterTable("G"); and CharacterTable("G") mod p;, where G is the name of the group, e.g. PSL(2,19) or M10. The corresponding decomposition matrix for a Brauer table can then be obtained by DecompositionMatrix.. We will use througout the GAP notation for conjugacy classes.
For and we have , there are cyclic subgroups of order , and in , and every cyclic subgroup of lies in a conjugate of such a subgroup. There are two conjugacy classes of elements of order and, if is an element of order prime to , the only conjugate of in is . All this follows from a result of Dickson [Hup67, Satz 8.27].
Separating the knwon HeLP-method and the new method we first list the results obtainable only using the HeLP-method.
Let be a prime, and set
If is neither nor , then elements in of order or are rationally conjugate to group elements.
If is neither nor and is of order 10, then either is rationally conjugate to a group element or it has the following partial augmentations:
if is rationally conjugate to an element in 5a, or
if is rationally conjugate to an element in 5b. The conjugacy classes are listed in a way, such that squares of elements in 10b are laying in 5a.
We will use the representations given in [Her07] and explicitly proved in [Mar14, Lemma 1.2], i.e.: If is an element of order and is an element of order in , then there is a primitive -th root of unity and a primitive -th root of unity such that for every there exists a -modular representation of with character such that
Let . If is of order 4, then by [Her07, Proposition 6.5]. Thus and is rationally conjugate to a group element.
Assume is of order 9. Then by [Her07, Proposition 6.5]. Let and 9c be the conjugacy classes of elements of order in such that if , we have and . Then
Let be a primitive complex -th root of unity such that Since and is real valued, we get with So
Using as a -basis of (cf. [Neu99, Chapter 1, (10.2) Proposition]) this gives
Combining each of this possibilities with , we get
Thus is rationally conjugate to a group element. This is also a consequence of [Mar14, Proposition 1].
Now assume is of order 12, then contains an element of order 12 by Proposition 2.1. So let be the conjugacy classes with potentially non-vanishing partial augmentations for . Let be a primitive 12-th root of unity such that
We will use as a -basis of . (This is a basis since , where denotes Euler’s totient function, and and .) We have
Furthermore, and Thus, as has only real values, with Hence, using Table 1, we obtain Using again as a basis of this gives
Proceeding the same way we have , and with So, by Table 1 and , we get . Comparing coefficients of gives
Applying the same for we obtain , and with
So, by Table 1, and , we get
As the first three possibilities would give , contradicting (2), only the last two remain and give
In the same way , , and (note that must not only be real, but even rational, as has only rational values). Thus giving
Now subtracting (1) from (6) gives while subtracting (4) from (5) gives . Thus . Then subtracting (1) from (3) gives , so Now multiplying (1) by 2 and subtracting it from (4) gives Using (1) and (2) this leaves only the trivial possibilities .
For part b) assume , is of order 10 and let be a primitive -th root of unity s.t. . Assume further that is rationally conjugate to an element in 5a.
Furthermore, and As has only real values, we get Thus
Using as a -basis of we obtain
In the same way we get , and with We have Hence
Combining these equations with the equations obtained above we get
If is rationally conjugate to an element in 5b, then replacing every with and doing the same computations as above gives the result. ∎
Proof of Theorem 1.
By Proposition 2.1, to obtain the Zassenhaus Conjecture for only elements of order and in need to be checked and these are rationally conjugate to group elements by Lemma 2.2. So from now on let . Then by Proposition 2.1 only elements of order 9 and 10 need to be checked, but elements of order 9 are already handled in Lemma 2.2. So assume is of order 10 and not rationally conjugate to a group element. If is not rationally conjugate to an element of , then also is not rationally conjugate to an element in . Furthermore, if is rationally conjugate to an element in 5a, then is rationally conjugate to an element in 5b. So we may assume that is conjugate to an element in 5a and by Lemma 2.2 we get
Let and be representations affording the characters and given in Table 3. We compute the eigenvalues of and using the character table in the way demostrated above: We have and . Since we obtain
Moreover and . Since we get