Zassenhaus Conjecture on torsion units holds
for with a Fermat or Mersenne prime
Abstract.
H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring of a finite group is conjugate in the rational group algebra to an element of the form with . Though known for some series of solvable groups, the conjecture has been proved only for thirteen nonabelian simple groups. We prove the Zassenhaus Conjecture for the groups , where is a Fermat or Mersenne prime. This increases the list of nonabelian simple groups for which the conjecture is known by probably infinitely many, but at least by , groups. Our result is an easy consequence of known results and our main theorem which states that the Zassenhaus Conjecture holds for a unit in of order coprime with , for some prime power .
2010 Mathematics Subject Classification:
16U60, 16S341. Introduction
One of the most famous open problems regarding the unit group of an integral group ring of a finite group is the Zassenhaus Conjecture which was stated by H.J. Zassenhaus [Zas74]:
Zassenhaus Conjecture^{1}^{1}1After this paper was submitted a metabelian counterexample to the Zassenhaus Conjecture was announced in [EM17]. Still no simple counterexample is known.: If is a finite group and is a unit of finite order in the integral group ring , then there exists a unit in the rational group algebra and an element such that .
If for a given such and exist, one says that and are rationally conjugate. The Zassenhaus Conjecture found much attention and was proved for many series of solvable groups, e.g. for nilpotent groups [Wei91], groups possessing a normal Sylow subgroup with abelian complement [Her06] or cyclicbyabelian groups [CMdR13]. Regarding nonsolvable groups, however, the conjecture is only known for very few groups. The proofs of the results for solvable groups mentioned above often argue by induction on the order of the group. In this way one may assume that the conjecture holds for proper quotients of the original group. The first step in a similar argument for nonsolvable groups should consist in proving the conjecture for simple groups. Although this has been studied by some authors, see e.g. [LP89, Her07, Her08, BKL08, Sal13, BM17, BC17], the conjecture is still only known for exactly thirteen nonabelian simple groups all being isomorphic to some for some particular small prime power (see [BM18, Theorem C] for an overview). Our aim in this paper is to extend this knowledge by proving the following theorem.
Theorem 1.1.
Let for some prime power . Then any torsion unit of of order coprime with is rationally conjugate to an element of .
We prove this result employing a variation of a well known method which uses characters of a finite group to obtain restrictions on the possible torsion units in . The idea of the method was introduced for ordinary characters by Luthar and Passi [LP89] and extended to Brauer characters by Hertweck [Her07]. Today this method is often called the HeLP (HertweckLutharPassi) Method. In fact to prove our results we do not use the HeLP Method in the classical sense, since this would imply too many case distinctions. For this reason we vary the method in a way suitable for the character theory of . Theorem 1.1 can be regarded as a generalization of [Mar16, Theorem 1].
As a direct application of Theorem 1.1 and known facts about the units of collected in Theorem 2.2, we obtain the result which gives name to this paper:
Theorem 1.2.
Let be a Fermat or Mersenne prime. Then the Zassenhaus Conjecture holds for .
This result increases the number of simple groups for which the Zassenhaus Conjecture is known from thirteen to sixtytwo: The groups with or one of the four known Fermat primes different from 3 or one of the fortynine known Mersenne primes different from 3 [Calb]. Actually, Theorem 1.2 proves the conjecture for probably infinitely many simple groups because, based on heuristic evidences, it has been conjectured that there are infinitely many Mersenne primes [Cala]. Lenstra, Pomerance and Wagstaff have proposed independently a conjecture on the growth of the number of Mersenne primes smaller than a given integer [Pom81, Wag83].
It has been shown in [dRS17] that a result as in Theorem 1.1 can not be achieved using solemnly the HeLP Method if the unit has order , where is coprime with and a prime bigger than . Looking on the orders of elements in , cf. Theorem 2.2, one should not expect a better result for the Zassenhaus Conjecture for when applying only this method. Thus, as so often in Arithmetics and Group Theory, the prime behaves very differently than the other primes.
2. Preliminaries
Let be a finite group. If , then denotes the order of , the cyclic group generated by is denoted by and denotes the conjugacy class of in . If is a ring then denotes the group ring of with coefficients in . Denote by the group of normalized units (i.e units of augmentation ) in . As mentioned above, we say that two elements of are rationally conjugate if they are conjugate in the units of .
The main notion to study rational conjugacy of torsion units in are the so called partial augmentations. If is an element of a group ring , with each , then the partial augmentation of at is defined as
The relevance of partial augmentations for the study of the Zassenhaus Conjecture is provided by a result of Marciniak, Ritter, Sehgal and Weiss. The following theorem states this result and collects some known information about partial augmentations.
Theorem 2.1.
Let be a finite group and let be an element of order in .

[MRSW87, Theorem 2.5] is rationally conjugate to element in if and only if for all and all divisors of .

[JdR16, Proposition 1.5.1] (BermanHigman Theorem) If then .

[Her07, Theorem 2.3] If then divides .

[Her07, Theorem 3.2] Let be a prime not dividing and let be a Brauer character of associated to a modular representation for a suitable modular system . Then extends to a Brauer character defined on the regular torsion units of , associated to the natural algebra homomorphism . Moreover, if are representatives of the regular conjugacy classes of then
(2.1)
We collect the group theoretical properties of and its integral group ring relevant for us.
Theorem 2.2.
Let where for some prime and let .

[Hup67, Hauptsatz 8.27] The following properties hold.

The order of is .

The orders of elements in are exactly and the divisors of and .

Two cyclic subgroups of are conjugate in if and only if they have the same order.

If with coprime with and multiple of then is conjugate in to an element of and the only elements of conjugate to in are and . In particular a conjugacy class of elements of order coprime with is a real conjugacy class.


If is a torsion element of of order coprime with , is root of unity in an arbitrary field and is an representation of then and have the same multiplicity as eigenvalues of . This follows from (1) and the formulas for multiplicities of eigenvalues of torsion units as presented in [Her07, Section 4].

[Mar16, Lemma 1.2] Let be a positive integer coprime with and let be an element of order . There exists a primitive th root of unity in a field of characteristic such that for every positive integer , there is a modular representation of of degree such that
We denote by the Brauer character associated with .
As mentioned in the introduction, we actually do not use the HeLP Method in its classical setting. We neither compute many inequalities involving traces as for example in the proofs of [Her07, Proposition 6.5] or [BKL08, Mar16], since these formulas turn out to be too complicated in our setting. Nor do we apply the standard equations obtained from character values on one side and possible eigenvalues on the other side as e.g. in the proofs of [Her07, Propositions 6.4, 6.7], [Her08] or [BM17, Lemma 2.2], since there are too many possibilities for these possible eigenvalues. Still this second strategy is closer to our approach.
3. Number theoretical results
In this section we prove two number theoretical results which are essential for our arguments and might be of independent interest. Our first proof of Proposition 3.2 below was very long. We include a proof which was given to us by Hendrik Lenstra. We are very thankful to him for his simple and nice proof.
For a prime integer and a nonzero integer let denote the valuation of at , i.e. the maximal nonnegative integer with . If, moreover, then denotes a complex primitive th root of unity and denotes the th cyclotomic polynomial, i.e. the minimal polynomial of over .
Lemma 3.1.
If and are positive integers and is a prime integer then .
Proof.
We argue by induction on . Suppose first that and let denote the set of primitive th roots of unity. Then is a root of for every and hence divides in . Therefore
Suppose that and assume that the lemma holds with replaced by . Then . As and is a primitive th root of unity, we have . ∎
Proposition 3.2.
Let be a positive integer. Let be integers and for every positive integer set
Let be a divisor of such that for every prime power dividing with . Then .
Proof.
Let and consider the polynomial . We can take , so that . By hypothesis, for every prime and every positive integer with dividing we have , or equivalently divides in . Thus divides in . Therefore . By Lemma 3.1, each belongs to . As we deduce that , as desired. ∎
For a positive integer and a subfield of , let denote a set of representatives of equivalence classes of the following equivalence relation defined on :
Corollary 3.3.
Let be a positive integer, let be a subfield of and let be the ring of integers of . For every let be an integer and for every integer define
Let be a divisor of such that for every prime power dividing with . Then .
Proof.
Apply Proposition 3.2 to the integers with denoting the class in containing . ∎
In the remainder of this section we reserve the letter to denote positive prime integers.
We now introduce some notation for a positive integer which will be fixed throughout. First we set
If moreover then we set
Next lemma collects two elementary properties involving this notation whose proofs are direct consequences of the definitions.
Lemma 3.4.
Let be a prime dividing and let . Then the following conditions hold:

If then .

Let such that . If divides both and then .
For integers and we define the following equivalence relation on :
We denote by a set of representatives of these equivalence classes. Without loss of generality one may assume that .
In the remainder of the section we assume that is odd. For and integers let
Moreover, is the maximal real subfield of and is the ring of integers of .
Let
In the following proposition we prove that is a basis of . For and , we use
We denote by the number theoretical Möbius function.
Proposition 3.5.
Let be a positive odd integer. Then

is a basis of and in particular, a basis of .

If and then
Proof.
It is easy to see that . Thus it is enough to prove the following equality
Actually we will show , which implies the desired expression of . Indeed, for every let denote the th part of , i.e. is a primitive th root of unity and . Let be the set of tuples satisfying for every . For every let given by
Then is the set of elements in satisfying . From
we obtain . Therefore
∎
4. Proof of Theorem 1.1
In this section we prove Theorem 1.1. In the remainder, set with a prime. Our goal is to prove that any element of order in , where is greater than and coprime with , is rationally conjugate to an element of . By Theorem 2.2.(3) we may also assume that is not a prime power.
As the order of is fixed throughout, we simplify the notation of the previous section by setting
We argue by induction on . So we assume that is rationally conjugate to an element of for every divisor of with .
We will use the representations and Brauer characters introduced in Theorem 2.2.(4). As usual in modular representation theory, a bijection between the complex roots of unity of order coprime with and the roots of unity of the same order in a field of characteristic has been fixed a priori. In this sense we will identify the eigenvalues of and the summands in . Since units of prime order in are rationally conjugate to elements of by Theorem 2.2.(3), we know that the kernel of on is trivial and hence has order . As the values of on regular elements of are real, by Theorem 2.2.(1) and Theorem 2.1.(4), the set of eigenvalues of is closed under taking inverses (counting multiplicities). Therefore, is conjugate to for a suitable primitive th root of unity . Hence by Theorem 2.2 there exists an element of order such that and are conjugate. From now on we abuse the notation and consider both as a primitive th root of unity in a field of characteristic and as a complex primitive th root of unity. Then for any positive integer we have that
(4.1) 
and for every integer we have
(4.2) 
The element and the primitive th root of unity will be fixed throughout.
By Theorem 2.2.(1), defines a bijection from to the set of conjugacy classes of formed by elements of order dividing . For an integer (or ) we set
By Theorem 2.1, is rationally conjugate to an element of if and only if for every .
Lemma 4.1.
is rationally conjugate to if and only if
(4.3) 
Proof.
If is rationally conjugate to , then and for any . Therefore (4.3) holds. Conversely, assume that (4.3) holds. For of order dividing let . Then and . As the Vandermonde matrix is invertible we deduce that for every . So for every and . As we are assuming that if is a divisor of different from then is rationally conjugate to an element of , we also have for every . Thus is rationally conjugate to an element of by Theorem 2.1.(1). Then and therefore is conjugate to in . We conclude that and are rationally conjugate. ∎
By Lemma 4.1, in order to achieve our goal it is enough to prove (4.3). We argue by contradiction, so suppose that for some positive integer which we assume to be minimal with this property. Observe that if and is an integer such that , then there exists such that and applying to the equation we obtain . This implies that divides . Note that by our choice of and hence . Moreover, because as the augmentation of is 1.
We claim that
(4.4) 
Indeed, for any let if and otherwise. Then for any integer we have . Therefore, applying Corollary 3.3 for , and , the claim follows.
Combining this with (4.4) and the minimality of , we obtain . Furthermore, , as . Therefore
(4.6) 
and
(4.7) 
The bulk of our argument relies on an analysis of the eigenvalues of and the induction hypothesis on and . More precisely, we will use (4.6) and (4.7) to obtain a contradiction by comparing the eigenvalues of and . Of course we do not know the eigenvalues of the latter but we know the eigenvalues of each . Moreover, if is a divisor of with then is rationally conjugate to an element of . Then , and are conjugate in , for a suitable field , and as is injective on and is conjugate to an element of we conclude that is conjugate to . Thus we know the eigenvalues of . This has consequences for the eigenvalues of .
To be more precise we fix (with repetitions if needed) such that the eigenvalues of with multiplicities are . This is possible by the last statement of Theorem 2.2.(1). By the above paragraph, if with then the lists and represent the same elements in , up to ordering, and hence and represent the same elements of , up to ordering. We express this by writing
This provides restrictions on , and the .
Moreover, and are the coefficients of in the expression in the basis of and , respectively. By (4.5) and Proposition 3.5 we obtain for every that
(4.8) 
and so
(4.9) 
Lemma 4.2.

If for some then is the smallest prime dividing and for every with .

If then is not divisible by any prime greater than .
Proof.
Let denote the smallest prime dividing .
(1) Suppose that . Then . As we deduce that for some . Therefore and for every with we have . Hence .
(2) Suppose that is a prime divisor of with . Then and therefore, by (1), for every . Thus, by (4.7) and (4.9) and ignoring the signs provided by the and , it is enough to show that for at most two ’s and for at most two ’s, since by assumption , i.e. . Observe that if then and hence is multiple of . Moreover, if with then . Therefore unless . As and we have , the lemma follows. ∎
For a nonzero integer let denote the number of prime divisors of . We obtain an upper bound for in terms of .
Lemma 4.3.
For every we have
Moreover if for every then
Proof.
Using (4.9), and ignoring the sings given by and , it is enough to prove that
Observe that for at most one by Lemma 4.2.(1). Recall that . Thus the lemma is a consequence of the following inequalities for every dividing :
since the number of divisors of is and if for some this provides an additional . We prove the second inequality, only using that . This implies the first inequality by applying the second one to .
For a fixed dividing let . By changing the sign of some ’s, we may assume without loss of generality that if then . Thus, if then . We claim that if then . Indeed, let be prime divisor of . If then , so . If then and so also . Otherwise, i.e. if and , then divides both and and . Therefore , by Lemma 3.4.(2). As and there are at most two ’s with representing the same class in , we deduce that , as desired. ∎
We are ready to finish the proof of Theorem 1.1. Recall that we are arguing by contradiction and , and hence also , is odd.
By (4.7) and Lemma 4.3 we have and this has strong consequences on the possible values of . Indeed if then
a contradiction. Thus, if then and if then .
However, if then by Lemma 4.3 and hence for one . This implies, by Lemma 4.2.(1), that contradicting the assumptions that is not a prime power. Therefore . We deal with these cases separately using (4.8) and (4.9). Observe that if is a prime bigger than then for every and so also , since .
Assume that . Combining Lemma 4.2.(