Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

In this paper, we essentially finish the classification of those finite 2, 3-groups G having integral group rings with the multiplicative Jordan decomposition (MJD) property. If G is abelian or a Hamiltonian 2-group, then it is clear that ℤ[G] satisfies MJD. Thus, we need only consider the nonabelian case. Recall that the 2-groups with MJD were completely determined by Hales, Passi and Wilson, while the corresponding 3-groups were almost completely determined by the present authors. Thus, we are concerned here, for the most part, with groups whose order is divisible by 6. As it turns out, there are precisely three nonabelian 2, 3-groups, of order divisible by 6, with ℤ[G] satisfying MJD. These have orders 6, 12, and 24. In view of another result of Hales, Passi and Wilson, this completes a significant portion of the classification of all finite groups with MJD.

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ℤ[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ℤ[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 33 = 27.

Multiplicative Jordan decomposition in group rings and p-groups with all noncyclic subgroups normal

Nilpotent decomposition in integral group rings

Corrigendum to “The multiplicative Jordan decomposition in group rings, II” [J. Algebra 316 (1) (2007) 109–132

The multiplicative Jordan decomposition in group rings, II

Integral group rings with all central units trivial

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

Let G be a finite group and be the integral group ring of G. We denote by the group of normalized units of that is, the units which have augmentation 1, and by the group of normalized central units. Many articles have been written describing the groups and for certain groups G. In this work, we will describe the group of normalized central units of some integral group rings by applying the idea presented in an article by Ferraz and Simón to a wider variety of groups, and we will study some examples of groups that can be treated with this method: metacyclic groups of type some metacyclic p-groups; some metabelian p-groups and some generalized dihedral groups.

The Multiplicative Jordan Decomposition in Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)∖{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.

Let X be a finite group, and denote its integral group ring by ZX. A group basis of ZX is a subgroup Y of the group of units of ZX of augmentation 1 such that ZX = ZY and IXI = YI. An example of a finite group X is given such that ZX has a group basis which is not isomorphic to X. A main ingredient is the existence of a subgroup G of X which possesses a non-inner automorphism which becomes inner in the integral group ring ZG. The question whether a finite group X is determined by its integral group ring ZX is known as the 'isomorphism problem for integral group rings'. It was

A code over a group ring is defined to be a submodule of that group ring. For a code $C$ over a group ring $RG$, $C$ is said to be checkable if there is $v\\in RG$ such that {$C=\\{x\\in RG: xv=0\\}$}. In \\cite{r2}, Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring $RG$ is code-checkable if every ideal in $RG$ is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring $\\mathbb{F}G$, when $\\mathbb{F}$ is a finite field and $G$ is a finite abelian group, to be code-checkable. In this paper, we give some characterizations for code-checkable group rings for more general alphabet. For instance, a finite commutative group ring $RG$, with $R$ is semisimple, is code-checkable if and only if $G$ is $\\pi'$-by-cyclic $\\pi$; where $\\pi$ is the set of noninvertible primes in $R$. Also, under suitable conditions, $RG$ turns out to be code-checkable if and only if it is pseudo-morphic.

This paper is intended to give a survey of recent work on central units in integral group rings. For units in general, the definitive reference is the book by Sehgal (1993) while a survey paper by Jespers contains additional very recent results. Both of these sources contain results on central units (in fact, Jespers devotes a chapter to the topic), but our work complements theirs in two ways. Firstly, we describe some results contained in papers which were not available to the other authors. Secondly, we choose to emphasize some topics which are mentioned either very briefly or not at all in their work. Nevertheless, we acknowledge that there is considerable overlap, especially between our survey and that of Jespers, and would like to thank him for supplying us with a preprint.

Non-Abelian composition factors of finite groups with the CUT-property