Units Of Integral Group Rings Research Articles

Complete group of central units in some group rings over Z and Z [ θ p ]

In the present work, we contribute with new results that enabled us to give an explicit description of the groups of central units of integral group rings and group rings with coefficients in Z [ θ ] , for certain groups G, where θ is a p − primitive root of unity and p is a prime number. In particular, we find a set of generators for the case where G is the Heisenberg Group.

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

Constructing free products of cyclic subgroups inside the group of units of integral group rings

It has been proved in Janssens, Jespers, and Temmerman [Proc. Amer. Math. Soc. 145 (2017), pp. 2771–2783] that if h h is an element of prime order p p in a finite nilpotent group G G and u = h + ( h − 1 ) g h ^ ∈ Z G u=h+(h-1)g\widehat {h}\in \mathbb {Z}G , u ∉ G u\not \in G , then ⟨ u ∗ , u ⟩ ≈ C p ∗ C p \langle u^*,u\rangle \approx C_p\ast C_p . We offer a simple geometric approach to generalize this result.

Abelianization and fixed point properties of units in integral group rings

AbstractLet G be a finite group and the unit group of the integral group ring . We prove a unit theorem, namely, a characterization of when satisfies Kazhdan's property , both in terms of the finite group G and in terms of the simple components of the semisimple algebra . Furthermore, it is shown that for , this property is equivalent to the weaker property (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property , denoted . More precisely, it is described when all subgroups of finite index in have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups , where is an order in a finite‐dimensional semisimple ‐algebra D, and finite groups G, which have the so‐called cut property. For such groups G, we describe the simple epimorphic images of . The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups of . These groups are well understood except in the degenerate case of lower rank, that is, for with an order in a division algebra D with a finite number of units. In this setting, we determine Serre's property FA for and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ‐rank.

Central units of integral group rings of monomial groups

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .

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.

Structure of group rings and the group of units of integral group rings: an invitation

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $$\mathcal {U}(\mathbb {Z}G)$$ of the integral group ring $$\mathbb {Z}G$$ of a finite group G. These constructions rely on explicit constructions of units in $$\mathbb {Z}G$$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $$\mathbb {Q}G$$ . The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

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.

A novel public-key encryption scheme based on Bass cyclic units in integral group rings

In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non-commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.

On units in orders in 2-by-2 matrices over quaternion algebras with rational center

We generalize an algorithm established in earlier work [21] to compute finitely many generators for a subgroup of finite index of an arithmetic group acting properly discontinuously on hyperbolic space of dimension 2 and 3, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators of finite index subgroups of a discrete subgroup of Vahlen’s group, i.e. a group of 2-by-2 matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group rings and this new algorithm allows to handle unit groups of orders in 2-by-2 matrices over rational quaternion algebras. The rings investigated are part of the so-called exceptional components of a rational group algebra.

Some results on FC-hypercentral units of integral group rings

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .

A generalization of strongly monomial groups

Olivieri, del Río and Simón defined strongly monomial groups and a significant result proved by them is the explicit description of the simple components of the rational group algebra QG of a strongly monomial group G. In this paper, generalized strongly monomial groups are defined, which is a generalization of strongly monomial groups. Beside strongly monomial groups, an extensive list of important families of groups which are generalized strongly monomial is produced. A description of the simple components of QG, when G is a generalized strongly monomial group, is also provided. Furthermore, the work by Jespers, Olteanu, del Río and Van Gelder on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has also been generalized.

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.

Coherent groups of units of integral group rings and direct products of free groups

AbstractWe classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$, the group of units of the integral group ring of G, does not contain a direct product of two non-abelian free groups. This list of groups contains all the groups for which $\mathcal{U}({\mathbb Z} G)$ is coherent. This reduces the problem to classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$ is coherent to decide about the coherency of a finite list of groups of the form SLn(R), with R an order in a finite dimensional rational division algebra.

Subgroup Isomorphism Problem for units of integral group rings

Abstract The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V ⁢ ( ℤ ⁢ G ) ${\mathrm{V}(\mathbb{Z}G)}$ , the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However, the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We give a positive solution to the Subgroup Isomorphism Problem for C 4 × C 2 ${C_{4}\times C_{2}}$ . Moreover, we prove that if the Sylow 2-subgroup of G is a dihedral group, any 2-subgroup of V ⁢ ( ℤ ⁢ G ) ${\mathrm{V}(\mathbb{Z}G)}$ is isomorphic to a subgroup of G.

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).

Involutions and free pairs of bicyclic units in integral group rings of non-nilpotent groups

If ∗ : G → G is an involution on the finite group G, then ∗ extends to an involution on the integral group ring Z[G]. In this paper, we consider whether bicyclic units u ∈ Z[G] exist with the property that the group 〈u, u∗〉, generated by u and u∗, is free on the two generators. If this occurs, we say that (u, u∗) is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. The main result here is that if G is a non-nilpotent group, then for any involution, Z[G] contains a free bicyclic pair.

Describing units of integral group rings up to commensurability

We restrict the types of 2×2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient.

Central units of integral group rings

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the same rank if $G$ is a finite group such that $\Q G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\Q$. This allows us to prove that in this case the natural images of the Bass units of $\Z G$ generate a subgroup of finite index in $\U(\Z G)/\U(\Z G)'$.

Central units in integral group rings II

Central units in integral group rings II