Zassenhaus Conjecture Research Articles

Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity

Abstract In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ , where p is a prime number and where the orbit of $0$ is finite. For example, if $p=2$ and $0$ is periodic under $T^2+c$ with $c\in \mathbb {R}$ , we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in $\mathbb {Q}(y)[T]$ .

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple modular Lie algebras H(2;(1,n))(2) of dimension 3n+1−2 in characteristic p=3 do not have a solvable outer derivation algebra for all n≥1. For n=1 this recovers the known counterexample of psl3(F). We show that the outer derivation algebra of H(2;(1,n))(2) is isomorphic to (sl2(F)⋉V(2))⊕Fn−1, where V(2) is the natural representation of sl2(F). We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.

Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question

Abstract The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well-studied class of rational groups, and has received attention recently. In the first part of this paper, we give a complete classification of the Gruenberg–Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg–Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg–Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg–Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The prime graph question asks whether the Gruenberg–Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the first Zassenhaus conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the prime graph question for integral group rings for finite rational groups and most finite cut groups

From examples to methods: two cases from the study of units in integral group rings

In this article, we review the proofs of the first Zassenhaus Conjecture on conjugacy of torsion units in integral group rings for the alternating groups of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of these examples led to the development of two methods – the HeLP method and the lattice method. We exhibit these methods and summarize some results which were achieved using them. We then apply these methods to the study of the first Zassenhaus conjecture for the alternating group of degree 7 where only one critical case remains open for a full answer. Along the way we show in examples how recently obtained results can be combined with the methods presented and collect open problems some of which could be attacked using these methods.

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Using the Luthar-Passi method, we investigate the Zassenhaus and Kim- merle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held sim- ple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture. Let U(ZG) be the unit group of the integral group ring ZG of a finite group G. It is well known that U(ZG )= U(Z) × V (ZG), where V (ZG )= g∈G αgg ∈ U(ZG) | g∈G αg =1 ,α g ∈ Z.

Zassenhaus Conjecture on torsion units holds for PSL(2,p) with p a Fermat or Mersenne prime

H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring ZG of a finite group G is conjugate in the rational group algebra QG to an element of the form ±g with g∈G. This is known to be true for some series of solvable groups, but recently metabelian counterexamples have been constructed. The conjecture is still open for non-abelian simple groups and has only been proved for thirteen such groups. We prove the Zassenhaus Conjecture for the groups PSL(2,p), where p is a Fermat or Mersenne prime. This increases the list of non-abelian simple groups for which the conjecture is known by probably infinitely many, but at least by 50, 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 ZPSL(2,q) of order coprime with 2q, for some prime power q.

Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring ℤ ⁢ G {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra ℚ ⁢ G {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups SL ⁢ ( 2 , p ) {\mathrm{SL}(2,p)} and SL ⁢ ( 2 , p 2 ) {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if G = SL ⁢ ( 2 , p f ) {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of ℤ ⁢ G {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in ℚ ⁢ G {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of ℤ ⁢ G {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

Partial augmentations power property: A Zassenhaus Conjecture related problem

Zassenhaus conjectured that any unit of finite order in the integral group ring ZG of a finite group G is conjugate in the rational group algebra of G to an element in ±G. 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 ZG, 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 G. 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.

On the First Zassenhaus Conjecture and Direct Products

AbstractIn this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

An algorithm to construct candidates to counterexamples to the Zassenhaus Conjecture

Let G be a finite group, N a nilpotent normal subgroup of G and let V(ZG,N) denote the group formed by the units of the integral group ring ZG of G which map to the identity under the natural homomorphism ZG→Z(G/N). Sehgal asked whether any torsion element of V(ZG,N) is conjugate in the rational group algebra of G to an element of G. This is a special case of the Zassenhaus Conjecture.By results of Cliff and Weiss and Hertweck, Sehgal's Problem has a positive solution if N has at most one non-cyclic Sylow subgroup. We present some algorithms to study Sehgal's Problem when N has at most one non-abelian Sylow subgroup. They are based on the Cliff–Weiss inequalities introduced by the authors in a previous paper. With the help of these algorithms we obtain some positive answers to Sehgal's Problem and use them to show that for units in V(ZG,N) our method is strictly stronger than the well known HeLP Method. We then present a method to use the output of one of the algorithms to construct explicit metabelian groups which are candidates to a negative solution to Sehgal's Problem. Recently Eisele and Margolis showed that some of the examples proposed in this paper are indeed counterexamples to the Zassenhaus Conjecture. These are the first known counterexamples. Moreover, we prove that every metabelian negative solution of Sehgal's Problem satisfying some minimal conditions is given by our construction.

Cliff–Weiss inequalities and the Zassenhaus Conjecture

Let N be a nilpotent normal subgroup of the finite group G. Assume that u is a unit of finite order in the integral group ring ZG of G which maps to the identity under the linear extension of the natural homomorphism G→G/N. We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of u and apply this to the study of the Zassenhaus Conjecture. This conjecture states that any unit of finite order in ZG is conjugate in the rational group algebra of G to an element in ±G.

On the Prime Graph Question for Integral Group Rings of 4-Primary Groups II

In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly $4$ different primes is continued. We provide more details on the recently developed "lattice method" which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the "lattice method" is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex $3$ provided the Sylow $3$-subgroup is of order $3$.

On the difference between permutation polynomials

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d2−3d+4)2, then there is no complete mapping polynomial f in Fp[x] of degree d≥2. For arbitrary finite fields Fq, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f.Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over Fp, with p>(d2−3d+4)2, then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in Fq[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+xk is a permutation over Fq, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).

The Status of the Zassenhaus Conjecture for Small Groups

ABSTRACTWe identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP+ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the lattice method. In the end, the Zassenhaus conjecture is verified for all groups of order less than 144 and we give a list of all remaining cases among groups of orders 144 to 287.

Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

AbstractWe 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 PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any \(d\ge 2\) and any prime \(p>(d^2-3d+4)^2\) there is no complete mapping polynomial in \(\mathbb {F}_p[x]\) of degree d. For arbitrary finite fields \(\mathbb {F}_q\), we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if \(n<\lfloor q/2\rfloor \), then there is no complete mapping in \(\mathbb {F}_q[x]\) of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank \(n<\lfloor q/2\rfloor \) are from being complete, by studying value sets of \(f+x.\) We provide examples of complete mappings if \(n=\lfloor q/2\rfloor \), which shows that the above bound cannot be improved in general.

Torsion units for some almost simple groups

We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL(2, 11). Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL(2, 13).

Torsion units for a Ree group, Tits group and a Steinberg triality group

We investigate the Zassenhaus conjecture for the Steinberg triality group \({}^3D_4(2^3)\), Tits group \({}^2F_4(2)'\) and the Ree group \({}^2F_4(2)\). Consequently, we prove that the Prime Graph question is true for all three groups.

Torsion units for some projected special linear groups

In this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.