Prime Graph Question Research Articles

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

Orders of units in integral group rings and blocks of defect 1

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$, where $q$ is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including a new infinite series of such groups. Our methods are based on the understanding of blocks of cyclic defect and Young tableaux combinatorics.

An application of blocks to torsion units in group rings

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p p -block of a finite group G G , the normalized unit group of the integral group ring of G G contains an element of order p q pq if and only if so does G G for q q a prime different from p p . Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.

On the prime graph question for integral group rings of Conway simple groups

The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group G contains an element of order pq, for some primes p and q, also G contains an element of that order. We answer this question for the three Conway sporadic simple groups after reducing it to a combinatorial question about Young tableaux and Littlewood-Richardson coefficients. This finishes work of V. Bovdi, A. Konovalov and S. Linton.

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 Gruenberg–Kegel graph of integral group rings of finite groups

The prime graph question asks whether the Gruenberg–Kegel graph of an integral group ring [Formula: see text], i.e. the prime graph of the normalized unit group of [Formula: see text], coincides with that one of the group [Formula: see text]. In this note, we prove for finite groups [Formula: see text] a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups [Formula: see text] whose order is divisible by at most three primes and show that the Gruenberg–Kegel graph of such groups coincides with the prime graph of [Formula: see text].

On the prime graph question for integral group rings of 4-primary groups I

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to [Formula: see text] for [Formula: see text], establishing the Prime Graph Question for all groups where the only non-abelian composition factors are of the aforementioned form. Using this, we determine exactly how far the so-called HeLP method can take us for (almost simple) groups having an order divisible by at most four different primes.

On the prime graph question for almost simple groups with an alternating socle

Let [Formula: see text] be an almost simple group with socle [Formula: see text], the alternating group of degree [Formula: see text]. We prove that there is a unit of order [Formula: see text] in the integral group ring of [Formula: see text] if and only if there is an element of that order in [Formula: see text] provided [Formula: see text] and [Formula: see text] are primes greater than [Formula: see text]. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle [Formula: see text] if [Formula: see text].

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.

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.