Group Ring Theory

Let F C0 be the class of all finite groups, and for each non- negative integer n define by induction the group class F Cn+1 consisting of all groups G such that for every element x the factor group G/CG(h xi G) has the property F Cn. Thus F C1-groups are precisely groups with finite conjugacy classes, and the class F Cn obviously contains all finite groups and all nilpotent groups with class at most n. In this paper the known theory of F C-groups is taken as a model, and it is shown that many properties of F C-groups have an analogue in the class of F Cn-groups.

In this paper we study some properties of associate and presimplifiable rings. We give a characterization of the associate (resp., domainlike) pullback P of R1 → R3 ← R2 , where R1 and R2 are two presimplifiable (resp., domainlike) rings. We prove that R is presimplifiable ring if and only if the factor ring R/nil(R) is presimplifiable and the ideal nil(R) is presimplifiable. Then we investigate the associate and presimplifiable property of the dual rings R[x]/ x 2 and its modules through the base ring R and its modules.

In this article we extend some theorems in Homological Algebra. we show that if 0 → X n-(k+1) → X n-k → ... → X n-1 → X n → 0 is an exact sequence of zero sequence, then for every k ∈ N , there exist a natural homeomorphism ϕ k : (k+1) ). Also by using α-sequences, we define CH(A) = Im(A i A i+1 ) Ker(A i+1 A i+2 ) , and prove that, if T (A) is a additive exact functor, X is a α-complex, and if T is covariant in A, then for every n ∈ N , CH n (T (X)) ∼ = T (CH n (X)).

A signature ε = (p, q) dependent transposition anti-involution T ε ˜of real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced in [1]. In [2] we showed that, depending on the value of (p -q) mod 8, the map T ε ˜gives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [12]. We provide a full signature (p, q) dependent classification of the invariance groups G ε p,q of this product for p + q ≤ 9. The map T ε ĩs identified as the "star" map known [14] from the theory of (twisted) group algebras, where the Clifford algebra Cℓ p,q is seen as a twisted group ring R t [(Z 2 ) n ], n = p + q. We discuss and list important subgroups of stabilizer groups G p,q (f ) and their transversals in relation to generators of spinor spaces.

In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which has been obtained by using a Kostant type cohomology formula for gl m|n . In general, we can obtain in a combinatorial way a Weyl type formula for various highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair.

We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences preserving orientation. We prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. We also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces.

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U(ZG)) of the unit group U(ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ritter and Sehgal [4] did the same for finite groups G, giving generators which are a little more complicated. They also gave in [2] necessary and sufficient conditions for Z(U(ZG)) to be trivial; recall that the units ±G are called the trivial units. We first give a finite set of generators for a subgroup of finite index in Z(U(ZG)) when G is a finite nilpotent group. Next we consider an arbitrary finitely generated nilpotent group and prove that a central unit of ZG is a product of a trivial unit and a unit of ZT, where T is the torsion subgroup of G. As an application we obtain that the central units of ZG form a finitely generated group and we are able to give an explicit set of generators for a subgroup of finite index.

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 ZG 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 Z(U(ZG)) of the unit group U(ZG) in case G is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in Z(U(ZG)) for all finite strongly monomial groups G. We call these units generalized Bass units. Finally, we show that the commutator group U(ZG)/U(ZG) and Z(U(ZG)) have the same rank if G is a finite group such that QG 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 ZG generate a subgroup of finite index in U(ZG)/U(ZG) .

Let G be a finite group and U(ZG) the unit group of the integral group ring ZG. We prove a unit theorem, namely a characterization of when U(ZG) satisfies Kazhdan's property (T), both in terms of the finite group G and in terms of the simple components of the semisimple algebra QG. Furthermore, it is shown that for U(ZG) this property is equivalent to the weaker property FAb (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in U(ZG) have both finite abelianization and are not a non-trivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SLn(O), where O is an order in a finite dimensional semisimple Q-algebra D, and finite groups G which have the so-called cut property. For such groups G we describe the simple epimorphic images of QG. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups En(D) of SLn(D). These groups are well understood except in the degenerate case of lower rank, i.e. for SL2(O) with O an order in a division algebra D with a finite number of units. In this setting we determine Serre's property FA for E2(O) and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its Z-rank.

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

We present a survey of some recent results on problems posed by Sudarshan Sehgal. In this paper some new results are presented on the following problems posed by Sudarshan Sehgal in [43] . The integral group ring of a group G is denoted by ZG. Its unit group we denote by U(ZG), its group of normalized units by UI (ZG) and its center by Z(U(ZG» . Problem 17: Give a presentation by generators and relations for the unit group U(ZG) of the integral group ring ZG for some finite groups G . Problem 23: Give generators up t.o finite index for U(ZG) if G is a finite group. Problem 18: Find a good estimate for the index (U1 (7l.. G) : B) , where B is the group generated by the Bass cyclic units and the bicyclic units. Problem 19: Is the group generated by the bicylic units torsion-free? Problem 29: Suppose that G is a finite nilpotent group . Does G have a normal complement N in UI (Z G) , i.e., UI (ZG) = N >4 G? Problem 43: Let G be a finite group. Is Nu(zG)(G) = G Z(U(ZG))? In the late ...

In this paper, we introduce a new generalization of coherent rings using the Gorenstein projective dimension. Let n be a positive integer or n = 1. A ring R is called a left Gn-coherent ring in case every finitely generated submodule of finitely generated free left R-modules whose Gorenstein projective dimensionn¡1 is finitely presented. We characterize Gn-coherent rings in various ways, using Gn-flat, Gn-injective modules and cotorsion theory.

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra QG for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in QG, we describe a subgroup of finite index in the group of units U (ZG) of the integral group ring ZG that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

We investigate how Legendre G-array pairs are related to several different perfect binary G-array families. In particular we study the relations between Legendre G-array pairs, Sidelnikov-Lempel-Cohn-Eastman Z q-1 -arrays, Yamada-Pott G-array pairs, Ding-Helleseth-Martinsen Z 2 × Z m p -arrays, Yamada Z (q-1)/2 -arrays, Szekeres Z m p -array pairs, Paley Z m p -array pairs, and Baumert Z m 1 p 1 × Z m 2 p 2 -array pairs. Our work also solves one of the two open problems posed in Ding [J. Combin. Des. 16 (2008), 164-171]. Moreover, we provide several computer search based existence and non-existence results regarding Legendre Z n -array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre Z 57 -array pair.

We investigate how Legendre ‐array pairs are related to several different perfect binary ‐array families. In particular we study the relations between Legendre ‐array pairs, Sidelnikov‐Lempel‐Cohn‐Eastman ‐arrays, Yamada‐Pott ‐array pairs, Ding‐Helleseth‐Martinsen ‐arrays, Yamada ‐arrays, Szekeres ‐array pairs, Paley ‐array pairs, and Baumert ‐array pairs. Our work also solves one of the two open problems posed by Ding. Moreover, we provide several computer search‐based existence and nonexistence results regarding Legendre ‐array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre ‐array pair.

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to projective representations of a finite group (Abelian), $G$ say, such that the corresponding twisted group ring has minimal center. The latter then translates to the fact that the corresponding 2-cocycle allows a minimal (none!) number of ray classes and this forces a decomposition of $G$ in cyclic components in a suitable way, cf. Zmud, M., Symplectic geometries and projective representations of finite Abelian groups, (Russian) Mat. Sb. (N.S.) 87(129) (1972), 3--17.. In this small paper, I will provide a way to represent an Abelian Group Clifford Algebra using a matrix, and then give a way to calculate whether or not the center is trivial.

Let $\mathfrak{R}$ be an associative ring graded by left cancellative monoid $\mathsf{S}$, and $e$ the neutral element of $\mathsf{S}$. We study the following problem: if $\mathfrak{R}_e$ is nil, then is $\mathfrak{R}$ nil/nilpotent? We have proved that if $\mathfrak{R}_e$ is nil (of bounded index) and $\mathsf{f}$- commutative, then $\mathfrak{R}$ is nil (of bounded index). Later, we have shown that $\mathfrak{R}_e$ being nilpotent implies $\mathfrak{R}$ is nilpotent. Consequently, we have exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furthermore, we have exhibited relations between graded rings and the problems of Köthe and Kurosh-Levitzki. We have proved that graded rings and $\mathsf{f}$-commutative rings provide positive solutions to these problems.

This paper focuses on the study of Him - groups. A Home - group is the non associative generalization of the classical group G, whose associativity and unitality are twisted by a compatible bijective map. We present more properties of Him - groups, Home - subgroups, Home - normal subgroups, Him - quotient groups and Home - group homomorphisms with examples. We prove the Zassenhaus Butterfly Lemma of Him - groups as a major result of this paper.

As an application of the above inequality, we obtain an interpolation between quasi-arithmetic means in the light of operator superquadratic functions. We provide re nements of known results. For example, the following interpolation between quasi-arithmetic means is proven: Let ψ,φ ∈ C(J) be strictly monotone functions and φ,ψ ≥ 0. If φ−1 ≥ 0 is operator superquadratic and ψ−1 ≤ 0 is operator subquadratic, then Mφ(A; Φ) ≤ Mφ(A; Φ) + n ∑

The article presents the analysis of the linear complexity of periodic q-ary sequences when changing k of their terms per period. Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. There has been obtained a recurrence relation and an estimate of the change in the linear complexity of these sequences, where q is a primitive root modulo equal to the period of the sequence. It can be inferred from the results that the linear complexity of these sequences does not sign ificantly decrease when k is less than half the period. The study summarizes the results for the binary case obtained earlier.

Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of this 2-category are the crucial new ingredient; they are the icons of the title. These can be thought of as "the oplax natural transformations whose components are identities", but we shall also give a more elementary description. We describe some properties of these icons, and give applications to monoidal categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and to bundles of bicategories. * The support of the Australian Research Council and DETYA is gratefully acknowledged.

The group ring RG of a group G over a ring R (with identity \(R)) is a separable algebra over its center if and only if the following conditions hold: (a) R is a separable algebra over its center; (b) the center of G has finite index in G: (c) the commutator subgroup G' of G has finite order m and m\(R) is invertiblein R. 1. Introduction. If F is a commutative ring and G a finite group of order n, then the group ring RG is a separable F-algebra if and only if «1(F) is invertible in F (1(F) is the identity of F). This is a well-known generalization of Maschke's theorem, a proof of which may be based on the ideas in [3, p. 41]. A ring is an Azumaya algebra if it is separable over its center. In particular, if n\(R) is invertible in F, then RG is an Azumaya algebra. This condition is not necessary however. For example, if G is any abelian group and F any commutative ring then RG is an Azumaya algebra. A less trivial example was provided by C. T. C. Wall in . He showed that RG is a direct sum of matrix rings over commutative rings when F is a ring of /?-adic integers and G is a semidirect product of a finite cyclic group of order not divisible by p and a finite, abelian /?-group. In particular, RG is an Azumaya algebra in this case. The purpose of this paper is to prove the following theorem which determines which group rings RG are Azumaya algebras. Theorem I. Let R be a ring with identity 1(F) and let G be a group. The group ring RG is an Azumaya algebra if and only if the following conditions are satisfied: (a) F is an Azumaya algebra; (b) the center of G has finite index in G; (c) the commutator subgroup G' of G has finite order m and m\(R) is invertible in R. It seems worthwhile to explicitly state the theorem in the classical case for comparison with Maschke's theorem. Let F be a field and G a finite group. Then the group algebra FG is an Azumaya algebra if and only if the order of the commutator subgroup of G is invertible in F.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with S G ⊆ S being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring S G is a pure subring of S, equivalently, S G is a direct summand of S as an S G-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion S G ⊆ S is pure. It turns out that if S G ⊆ S is pure, then either the invariant ring S G is regular, or the group G is linearly reductive.

Use of “ring” in the determine of a planar ternary ring seems unjustified at first sight. In this paper we show that in Desargues affine plane, in certain condition, planar ternary ring (S, t) turns into usually associative ring (S, +, ·). So, in an affine plane A = (P, L, I) coordinatized from a coordinative system (O, I, OX, OY, OI) and bijection σ : (OI)→ S, determine a ternary operation t : S3 → S, which we call planar ternary operation. We prove that every coordinatizing affine plane A = (P, L, I) determine a planar ternary ring (S, t), where S is the set of coordinates of affine plane and t its planar ternary operation, and vise versa. Also we introduce the binary operation of addition and multiplication in S and underline relations a+ b = t(a, 1, b), a · b = t(a, b, 0). Since affine plane is fulfilled with the first Desargues axiom D1, related to the structure (S, +, ·) in that plane, considering some isomorphism imply that (S, +) is abelian group. In the following, we show t...

Within the quantum function algebra Fq[GLn], we study the subset Fq[GLn] — introduced in [Ga1] — of all elements of Fq[GLn] which are Z [ q, q ] –valued when paired with Uq(gln) , the unrestricted Z [ q, q ] –integral form of Uq(gln) introduced by De Concini, Kac and Procesi. In particular we yield a presentation of it by generators and relations, and a PBW-like theorem. Moreover, we give a direct proof that Fq [GLn] is a Hopf subalgebra of Fq[GLn], and that Fq[GLn] ∣∣ q=1 ∼= UZ(gln ∗) . We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from Fε[GLn] to F1[GLn] ∼= UZ(gln ∗) , also introduced in [Ga1]. The same analysis is done for Fq [SLn] and (as key step) for Fq [Mat n] .

We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that such functions are quasi-polynomials, which can be represented as alternative sums of Ehrhart quasi-polynomials associated with rational conic polytopes. In particular, we obtain generalizations of main theorems on difference dimension polynomials and their invariants to the case of weighted basic difference operators.

We consider generators of algebraic curvature tensors R which can be constructed by a Young symmetrization of product tensors U ⊗ w or w ⊗ U , where U and w are covariant tensors of order 3 and 1. We assume that U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). We show that the set S contains exactly one symmetry class S 0 ∈ S whose elements U ∈ S 0 can not play the role of generators of tensors R. The tensors U of all other symmetry classes from S \ {S 0 } can be used as generators for tensors R. Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 8 summands. In special cases these numbers can be reduced to the minimum 4. If this minimum occurs then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of alternating 2-tensor fields. We apply ideals and idempotents of group rings C[S r ] of symmetric groups S r , Young symmetrizers, discrete Fourier transforms and Littlewood-Richardson products. For symbolic calculations we used the Mathematica packages Ricci and PERMS.

We consider generators of algebraic covariant derivative curvature tensors R ′ which can be constructed by a Young symmetrization of product tensors W ⊗ U or U ⊗ W , where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ] and discrete Fourier transforms for symmetric groups S r . For symbolic calculations we used the Mathematica packages Ricci and PERMS.

Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S r ] of a symmetric group S r . If for a class of tensors T such a W is known, the elements of the orthogonal subspace W ⊥ of W within the dual space K[S r ] * of K[S r ] yield linear identities needed for a treatment of the term combination problem for the coordinates of the T . We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for S r to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS.

We consider generators of algebraic covariant derivative curvature tensors R ′ which can be constructed by a Young symmetrization of product tensors W ⊗ U or U ⊗ W , where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ] and discrete Fourier transforms for symmetric groups S r . For symbolic calculations we used the Mathematica packages Ricci and PERMS.

We consider generators of algebraic curvature tensors R which can be constructed by a Young symmetrization of product tensors U ⊗ w or w ⊗ U , where U and w are covariant tensors of order 3 and 1. We assume that U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). We show that the set S contains exactly one symmetry class S 0 ∈ S whose elements U ∈ S 0 can not play the role of generators of tensors R. The tensors U of all other symmetry classes from S \ {S 0 } can be used as generators for tensors R. Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 8 summands. In special cases these numbers can be reduced to the minimum 4. If this minimum occurs then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of alternating 2-tensor fields. We apply ideals and idempotents of group rings C[S r ] of symmetric groups S r , Young symmetrizers, discrete Fourier transforms and Littlewood-Richardson products. For symbolic calculations we used the Mathematica packages Ricci and PERMS.

Symmetry properties of r-times covariant tensors T can be de- scribed by certain linear subspaces W of the group ring K(Sr) of a symmet- ric group Sr. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W? of W within the dual space K(Sr) of K(Sr) yield linear identities needed for

The connection between computability and de…nability is one of the main themes in computable model theory. A decidable theory has a decidable model, that is, a model with a decidable elementary diagram. On the other hand, in a computable model only the atomic diagram must be decidable. For some nonisomorphic structures that are elementarily equivalent, we can use computable (in…nitary) sentences to describe different structures. In general, if two computable structures satisfy the same computable sentences, then they are isomorphic. Roughly speaking, computable formulas are L!1!-formulas with disjunctions and conjunctions over computably enumerable index sets. Let > 0 be a computable ordinal. A computable ( , resp.) formula is a computably enumerable disjunction (conjunction, resp.) of formulas 9u (x; u) (8u (x; u), resp.) where is computable ( , resp.) for some < . A computable 0 or 0 formula is a …nitary quanti…er-free formula. To show that our Harizanov was partially suppor...

We compute the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G. Next we construct a virtual basis of the group of central units of ZG for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QG. Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG, this for metacyclic groups G of the form G = C q m ⋊ C p n with p and q different primes and the cyclic group C p n of order p n acting faithfully on the cyclic group C q m of order q m .

We prove that a Dedekind domain R , graded by a nontrivial torsionfree abelian group, is either a twisted group ring k'[G] or a polynomial ring k[X], where k is a field and G is an abelian torsionfree rank one group. It follows that R is a Dedekind domain if and only if R is a principal ideal domain. We also investigate the case when R is graded by an arbitrary nontrivial torsionfree monoid.

Let G be a finite group, (ZG) the group of units of the integral group ring ZG and 1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively (ZG) for particular groups G. This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described (ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of (ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

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 ZG 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 Z(U (ZG)) of the unit group U (ZG) in case G is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in Z(U (ZG)) for all finite strongly monomial groups G. We call these units generalized Bass units. Finally, we show that the commutator group U (ZG)/U (ZG) ′ and Z(U (ZG)) have the same rank if G is a finite group such that QG 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 ZG generate a subgroup of finite index in U (ZG)/U (ZG) ′ .

For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if N U (G) = Z(U)G. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being nontrivial. Groups G for which class-preserving automorphisms are inner automorphisms, Outc(G) = 1, have the normalizer property. Recently, Herman and Li have shown that Outc(G) = 1 for a finite Blackburn group G. We show that Outc(G) = 1 for the members G of a few classes of metabelian groups, from which the Herman-Li result follows. Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group Z∞(U) of hypercentral units of U is contained in Z(U)G.