Partial Augmentations Power property:
A Zassenhaus Conjecture related problem
Zassenhaus conjectured that any unit of finite order in the integral group ring of a finite group is conjugate in the rational group algebra of to an element in . We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in , which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions.
We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of . Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.
Key words and phrases:Integral group ring, groups of units, Zassenhaus Conjecture, partial augmentation
1991 Mathematics Subject Classification:16U60, 16S34, 20C05, 20C10
1. Open problems on torsion units of integral group rings
The structure of the group of units of an integral group ring of a group has given rise to many interesting results and even more questions. In particular for finite , the units of finite order in were already studied by G. Higman, who showed that for abelian all are of the form for some [Higman1940]. Note that, denoting by the elements of whose coefficients sum up to , we have . So to study the unit group of it is sufficient to study .
For non-abelian already the smallest examples show that a result in the spirit of Higman is not valid and the strongest result one could hope for was put forward by H. J. Zassenhaus as a conjecture.
Zassenhaus Conjecture (ZC): [Zassenhaus] Let be a finite group and a unit in of finite order. Then there exists a unit in the rational group algebra of and an element such that .
If for such and exist we say that and are rationally conjugate. The Zassenhaus Conjecture has been proven for some classes of solvable groups, such as nilpotent groups [Weiss1991], cyclic-by-abelian groups [CaicedoMargolisdelRio2013], or groups which possess a normal Sylow subgroup with abelian complement [Hertweck2006]. It has also been proven for some particular non-solvable groups [LutharPassi1989, HertweckA6, 4primaryII, FermatMersenne].
Recently a metabelian counterexample to the Zassenhaus Conjecture has been obtained by Eisele and Margolis [EiseleMargolis]. It still seems almost hopeless to classify the groups for which the Zassenhaus Conjecture holds in wide classes of groups as e.g. solvable groups, or even metabelian group. Therefore also weaker versions of this conjecture are studied. For example Kimmerle proposed to study the following problem [Ari] which has been studied in [KimmerleMargolis17].
Kimmerle Problem, (KP) Given a finite group and an element of finite order in , is there a finite group containing as subgroup such that is conjugate in to an element of ?
Recall that the spectrum of a group is the set of orders of its torsion elements. The prime graph of has vertices labelled by the primes appearing as orders of torsion elements of and two vertices and are connected if and only if possesses an element of order . Clearly, if the Zassenhaus Conjecture has a positive solution for a group then and have the same spectrum and hence the same prime graph, too. This yields the following problems.
The Spectrum Problem (SP): (see e.g. [Sehgal1993, Problem 8]) Given a finite group , do and have the same spectrum?
The Prime Graph Question (PQ): (introduced in [Kimmerle2006]) Given a finite group , do and have the same prime graph?
The Spectrum Problem and the Prime Graph Question have been answered for much wider classes than the Zassenhaus Conjecture. The Spectrum Problem has a positive answer for solvable groups [HertweckOrders] and Frobenius groups [KimmerleKonovalov2016, Corollary 2.5]. The Prime Graph Question also for many sporadic simple groups (see e.g. [BovdiKonovalovConway]) or some infinite series of almost simple groups [4primaryI]. Moreover in contrast to the Zassenhaus Conjecture and the Spectrum Problem, for the Prime Graph Question a reduction theorem, to almost simple groups, is known [KimmerleKonovalov2016, Theorem 2.1]. The role of the prime graph for the structure of integral group rings is described in detail in [Kimmerle2006]. For an overview of the known results see also [BaechleKimmerleMargolisDFG]. Without putting assumptions on , the most far reaching result regarding the Spectrum Problem is a theorem by Cohn-Livingstone which states that the exponents of and coincide [CohnLivingstone].
In fact Cohn and Livingstone showed that certain sums of coefficients of torsion elements in satisfy some congruences [CohnLivingstone, Proposition 4.1]. Before we state these congruences denote by the sum of the coefficients of at , i.e.
for , with for each , and a subset of . If is a conjugacy class in then is called the partial augmentation of at . The importance of partial augmentations in the study of the Zassenhaus Conjecture and related questions was first realized by Marciniak, Ritter, Sehgal and Weiss who proved the following: A torsion element is rationally conjugate to an element of if and only if for every and any conjugacy class of [MarciniakRitterSehgalWeiss1987, Theorem 2.5].
For a positive integer denote by the set of elements of of order . Moreover let denote the set of all conjugacy classes in and for and an integer write if for . The Cohn-Livingstone congruences states that if is a unit of order , with prime, then for every integer we have
This has inspired A. Bovdi to pose the following question.
Bovdi Problem (BP): [BovdiBook, p. 26] (cf. also [ArtamonovBovdi1989, Problem 1.5]) Let be a torsion element of of prime power order . Is for every and ?
This question was studied for special classes of groups e.g. in [Juriaans1995] or [HertweckBrauer, Proposition 6.5]. Clearly as an intermediate step between the Zassenhaus Conjecture and the Spectrum Problem one could pose the following generalization of the question of Bovdi.
General Bovdi Problem (Gen-BP): Let be a torsion element of of order . Is for every ?
This is also stated as Problem 44 in [Sehgal1993]. It was studied for certain infinite groups in [Dokuchaev1992] and proved for metabelian non-necessarily finite groups in [DokuchaevSehgal1994, Corollary 1.4]. Let be a torsion unit of order . More evidence on (Gen-BP) is that it holds for , by the Berman-Higman Theorem, and if does not divide , by a theorem of Hertweck which states that if the order of elements in does not divide [HertweckBrauer, Theorem 2.3]. We will use these facts in the following without further mention.
A folklore congruence for , a prime , an integer and is the following:
(See [BovdiHertweck2008, Remark 6] for some references and [HeLPPaper, Proposition 3.1] for a proof for the case where which actually works in general.) This suggests to introduce the following property, in the spirit of (BP) and (Gen-BP).
The Partial Augmentation Power Property (PAP): We say that an element of satisfies the Partial Augmentation Power Property (PAP Property for short) if the following holds for every integer :
We say that satisfies the PAP Property if any torsion element of satisfies the PAP Property.
Note that each element of satisfies the PAP Property. Moreover, the partial augmentations of rationally conjugate units coincide. Thus the PAP Property holds for every unit rationally conjugate to an element of and, in particular, the Zassenhaus Conjecture implies the PAP Property.
In Section 2 we summarize the logical connections between the problems mentioned above. In Section LABEL:SectionPAP we prove the PAP Property for a particularly interesting class of units and give some practical features of the PAP Property. In Section LABEL:SectionPAPHeLP we analyze the implications and connections between the PAP Property and the HeLP Method. We show how these can be applied to the Zassenhaus Conjecture by proving the following theorem.
Let be a finite group with an abelian subgroup of prime index in . Assume that is generated by at most two elements. Then the Zassenhaus Conjecture holds for .
2. Connections between the problems
The logical connections between the problems are collected in the following proposition.
For a finite group the implications pictured in Figure LABEL:Logic hold.