The Projective Class Group of the Fundamental Group of a Surface is Trivial

Let D=F1 x F2 x... x Fn be a direct product of n free groups F1, F2, * , F* * , ox an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed, T an infinite cyclic group and F another free group. Let D x a T be the semidirect product of D and T with respect to a and (D x a T) x aXIdT F the semidirect product of D xa Tand F with respect to the automorphism x id T of D Xa T induced by a. We prove that the Whitehead group of (D xa, T) X 2xidT F and the projective class group of the integral group ring Z((D x a T) X aXidT F) are trivial. These results extend that of [3]. Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by kOZ(G). We recall the definition of semidirect product of groups and the definition of twisted group ring. For undefined terminologies used in the paper, we refer to [3] and [4]. Let oc be an automorphism of G and F a free group generated by {tA}. If w is a word in tA defining an element in F, we denote by Iwl the total exponent sum of the tA appearing in w. The semidirect product G xa F of G and F with respect to a is defined as follows: G x . F=GxF as sets and multiplication in G x . Fis given by (g, w)(g', w') = (go-lwl(g'), ww'), for any (g, w), (g', w') in G x F. In particular, if F is an infinite cyclic group T= (t) generated by t, we have the semidirect product G x a T of G and T with respect to oc. Let R be an associative ring with identity and oc an automorphism of R. Let F be a free group (or free semigroup) generated by {tA}. The otwisted group ring R,[F] of F over R is defined as follows: additively R,[F]=R[F], the group ring of F over R, so that its elements are finite linear combinations of elements in F with coefficients in R. Multiplication in R,[F] is given by (rw)(rIw')=roc-1I1(r')ww', for any rw, r'w' in R,[F]. In particular, if F is a free group (resp. free semigroup) generated by t, we Received by the editors May 25, 1973. AMS (MOS) subject classfiJcations (1970). Primary 13D15, 16A26, 18F25; Secondary 16A06, 16A54.

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

Picard groups and automorphism groups of integral group rings of metacyclic groups

Presenting an explicit descryption of unit group in the integral group ring of a given non-abelian group is a classical and open problem. Let S3 be a symmetric group of order 6 and C3 be a cyclic group of order 3. In this study, we firstly explore the commensurability in unit group of integral group ring ℤ(S3 × C3) by showing the existence of a subgroup as (F55 ⋊ F3) ⋊ (S3∗× C2) where Fρ denotes a free group of rank ρ. Later, we introduce an explicit structure of the unit group in ℤ(S3 × C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 where R = ℤ[ω] is the complex integral domain since ω is the primitive 3rd root of unity. At the end, we give a general method that determines the structure of the unit group of ℤ(G × C3) for an arbitrary group G depends on torsion-free normal complement V (G) of G in U(ℤ(G × C3)) in an implicit form. As a consequence, a conjecture which is found in [21] is solved.

Semi-clean group rings

Let p be an odd prime and \zeta_p be a primitive pth root of unity over \smallBbb{Q}. The Galois group G of K:=\smallBbb{Q}(\zeta_p) over \smallBbb{Q} is a cyclic group of order p-1. The integral group ring \smallBbb{Z}[G] contains the Stickelberger ideal S_p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S_p(q) obtained as reductions of S_p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S_p(p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B_{2k}, 1\le k\le (p-3)/2, which are divisible by p. We then develop methods to compute the generator polynomial of S_p(p). This gives rise to a new algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.

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

The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. In this note we examine and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers Z then R[G] is universally equivalent to Z[F] again with respect to an ap-propriate language.

Let G be a finite group and ZG its integral group ring. We show that if α is a non-trivial bicyclic unit of ZG, then there are bicyclic units β and γ of different types, such that 〈α, β〉 and 〈α, γ〉 are non-abelian free groups. In case that G is non-abelian of order coprime with 6, then we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ZG, such that for every positive integer m big enough, 〈u, v〉 is a free non-abelian group.

We show that PD4-complexes with free fundamental group are determined by their intersection pairings and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. The purpose of this article is to show that some of the basic properties of PD4-complexes with free fundamental group can be derived homologically, without reference to the topology of 4-manifolds or stabilization by connected sums, as used in (4, 8, 13). We also avoid explicit calculations of obstructions, relying instead on the easily verified fact that the 3-skeletons of the complexes considered have su‰ciently many self homotopy equivalences. In particular we give a new proof of the fact that such complexes are determined by their intersection pairings, and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. In the final section we consider briefly the classification (up to s-cobordism) of closed 4-manifolds with free fundamental group.

Let G = ⟨ a ⟩ ⋊ X G = \left \langle a \right \rangle \rtimes X where ⟨ a ⟩ \left \langle a \right \rangle is a cyclic group of order n , X n,X is an abelian group of order m m , and ( n , m ) = 1 (n,m) = 1 . We prove that if Z G \mathbb {Z}G is the integral group ring of G G and H H is a finite group of units of augmentation one of Z G \mathbb {Z}G , then there exists a rational unit γ \gamma such that H γ ⊆ G {H^\gamma } \subseteq G .

Describing the group of units in the integral group ring is a famous and classical open problem. Let S_3 and C_3 be the symmetric group of order 6 and a cyclic group of order 3, respectively. In this paper, a description of the units of the integral group ring Z(S_3×C_3) of the direct product group S_3×C_3 concerning a complex representation of degree two is given. As a result, a part of the conjecture which is introduced in (Low, 2008) and related to group rings over a complex integral domain is resolved using representation theory.

Let Gk be a cyclic group of order k, and let ZGk denote its group ring over the ring Z of rational integers. We denote by n(ZGk) the number of non-isomorphic indecomposable left ZGk-modules having finite Z-bases. In 1938 Diederichsen [2] proved that n(ZG,) is finite for p a rational prime, and gave an incorrect proof that n(ZG4) is infinite. Recently Roiter [6] and Troy [8] independently showed that n(ZG4) = 9. The present authors [4] have shown more generally that n(ZGP2) is finite for all primes p. The aim of this paper is to establish the somewhat surprising result that n(ZGP3) is infinite for each prime p. Several interesting consequences of this result should be pointed out. To begin with, note that GP3 is a homomorphic image of Gp, for each r ? 3, and so each ZGP3-module may be viewed also as a ZGpr-module. Therefore n(ZGpr) _ n(ZGP3), which shows that n(ZGpr) is infinite for each r > 3. Next, let Hp denote (for the moment) a p-Sylow subgroup of an arbitrary finite group G. The authors have shown [4] that if n(ZHp) is infinite for some p, then also n(ZG) is infinite. They proved furthermore that n(ZHp) is necessarily infinite if Hp is non-cyclic. Combining this result with that of the present paper, it follows that n(ZHp) is finite if and only if Hp is cyclic of order p or p2. Thus, the possible groups G for which n(ZG) is finite are small in number; in fact, if n(ZG) is finite, then for each p dividing the order of G each p-Sylow subgroup of G must be cyclic of order p or p2. Whether the converse is true is as yet unknown.t Let Z* denote the ring of p-adic integers in the p-adic completion of the rational field. Define n(Z*Gpr) to be the number of non-isomorphic indecomposable left Z*Gpr-modules having finite Z*-bases. In their previous paper [4], the authors have shown that n(ZGpr) is finite if and only if n(Z*Gpr)is finite. It is therefore sufficienttoprovethatn(Z*GP3) is infinite. Section 1 contains some general remarks about extensions of one direct

In this chapter we begin the detailed study of the SFC property for group rings of the form Z[F n ×Φ] where F n is the free group of rank n≥1 and Φ is finite. In the first instance we consider the rings Z[F n ×C m ] where C m is the cyclic group of order m.

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3, and Q8. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebra FCp is *-clean, where F is a field and Cp is a cyclic group of prime order p. In this article, we extend the above mentioned result from FCp to FqCpk, where Fq is a finite field and Cpk is a cyclic group of an odd prime power order pk. For the general case when G = Cn is cyclic group of order n, we also provide a necessary condition and a few sufficient conditions for FqCn to be *-clean.