Group of automorphisms of the ring ℤA4

A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H holds for G metabelian. Dade [9] showed there were counterexamples, even in the metabelian case, if Z were replaced by the family of all fields. Some mathematicians came to believe the problem was a kind of grammatical accident, that counterexamples for Z were surely there, if difficult to find. We ourselves began this investigation looking for counterexamples, but found that they were indeed very difficult to find, much more difficult than we had anticipated. Slowly we began to believe that at least some exciting positive results were possible. In this paper we answer the isomorphism problem for finite p-groups over the p-adic integers Z, = Zcp), and in a very strong way: In the normalized units (augmentation 1) of Z pG there is only one con&gay cZu.ss of groups of order ] G 1. This answers the isomorphism problem in the affirmative (for finite p-groups G, over Z or Zp) and in addition computes the entire Picard group [2] of the category of Z,G-modules in terms of automorphisms of G. Similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal Picard groups. We also treat more general coefficient domains: namely, integral domains S of characteristic 0 in which no (rational) prime divisor of the group order is invertible, for the SG isomorphism problem, and treat similar semilocal Dedekind domains for the Picard group computations. The Zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting (cf. Corollary 3 below). Over Z, we announce a positive answer to the isomorphism

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

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.

Let H be a generalized dihedral, semi-dihedral, quaternion, or modular group, and let A = (u, v, w) be a product of three odd order cyclic groups, with (|v|,|w|) = 1. For R a semi-local Dedekind domain of characteristic 0 in which no prime divisor of |H|.|A| is invertible, we prove that there is a semi-direct product G = H × A such that the group ring RG has an exceptional automorphism, i.e. provides a counter-example to a well-known conjecture of Zassenhaus on automorphisms of group rings

Projective limits of group rings

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).

The first object is a survey on the isomorphism problem of integral group rings and the Zassenhaus conjectures related to this question. Then it is shown how automorphisms of character tables may be used to get information about automorphisms of integral group rings. This permits the proof of one of the Zassenhaus conjectures for series of simple or almost simple groups. Finally a short proof is given that {1, 2, 3}-characters determine a finite group up to isomorphism. Introduction R. Brauer in his famous lectures in 1963 on representations of finite groups posed more than 40 problems which have had a big influence on this topic since then [7]. Many of these problems concern the question of which properties of a finite group G are reflected by its character table. The basic problem is to determine what kind of information about G additional to its character table is needed in order to determine G up to isomorphism. We shall consider two aspects of these questions. The first one is the isomorphism problem of integral group rings of finite groups. This is an old and still open problem. It was considered for the first time in G. Higman's thesis in 1939 [38, p. 100]. The question is whether ℤ G ≅ ℤ H implies that G ≅ H . It is a result of G. Glauberman that finite groups G and H with isomorphic integral group rings ℤ G and ℤ H have the same character table [16, (3.17)]. Thus the isomorphism problem fits well into the context of Brauer's problems. Note that K. W. Roggenkamp and A. Zimmermann recently constructed two non-isomorphic infinite polycyclic groups G and H whose integral group rings are Morita equivalent.

The object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

Suppose that a group G has a normal subgroup C where C and G/C are cyclic of relatively prime orders.

Suppose that a group G G has a normal subgroup C C where C C and G / C G/C are cyclic of relatively prime orders. Then any torsion unit in Z G ZG is rationally conjugate to a trivial unit.

In recent years several new restrictions on integral partial augmentations for torsion units of \(\mathbb {Z} G\) have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups G. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 – the non-split covering group of S4, and one of order 72 of isomorphism type (C3 × C3) ⋊ D8. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of \(\mathbb {Q} G\) that are not conjugate to elements of \(\mathbb {Z} G\).

Zassenhaus Conjecture for torsion units states that every aug- mentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups. In this paper G is a finite group and RG denotes the group ring of G with coefficients in a ringR. The units of RG of augmentation one are usually called normalized units. In the 1960s Hans Zassenhaus established a series of conjectures about the finite subgroups of normalized units of ZG. Namely he conjectured that every finite group of normalized units of ZG is conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units with the same cardinality as G is usually denoted (ZC2). These conjectures have important consequences. For example, a positive solution of (ZC2) implies a positive solution for the Isomorphism and Automorphism Problems (see (Seh93) for details). The most celebrated positive result for Zassenhaus Conjectures is due to Weiss (Wei91) who proved (ZC3) for nilpotent groups. However Roggenkamp and Scott founded a counterexample to the Automorphism Problem, and henceforth to (ZC2) (see (Rog91) and (Kli91)). Later Hertweck (Her01) provided a counterexample to the Isomorphism Problem. The only conjecture of Zassenhaus that is still up is the version for cyclic sub- groups namely:

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ℤG of a finite group G conjugates to a group element within the rational group algebra ℚG. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ℤA6.

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.

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