Unit group of integral group ring ℤ(G × C3)

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.

The main object of this paper is the determination of all the possible groups whose group of isomorphisms is either the symmetric group of order 6 or the synimetric group of order 24. We shall also determine the infinite system of groups whose group of cogredient isomorphisms is the former of these two symmetric groups. It will be proved that this system includes one and only one group (which is not the direct product of an abelian and a non-abelian group) for every power of 2. It is well known that every simple isomorphism of a group G with itself may be obtained by transforming G by means of operators that transform it into itself.t In what follows we shall generally employ this method of making G simply isomorphic with itself. In a few cases it will be convenient to employ two special methods, which 'we proceed to explain. The first of these two methods may be employed when G contains a subgroup H' which is composed entirely of operators which are selfconjugate under G and which is also simply isomorphic to a quotient group of G with respect to a selfconjugate subgroup which includes H'. In this case we may evidently multiply all of the operators of each one of the various divisions of G with respect to this quotient group by the corresponding operator of H' and thus obtain a simple isomorphisin of G with itself.-To illustrate this method we may employ the direct product G12 of' the symmetric group of order 6 anid an operator s1 of order two. If 'we multiply each of the six operators of G12 which are not contained in its cyclical subgroup of order 6 by s1 we obtain a simple isomorphism of G12 with itself. It is evident that this isomorphism corresponds to the selfconjugate operator of order two in the group of isomorphisms of G12 t It is important to observe that any operator t1 of the group of isomorphisms of G which is obtailned in this manner is selfconjugate under this group of isomorphisms whenever H' is composed of characteristic operators

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

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].

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

Using nonrigid group theory, the full nonrigid (f‐NRG) group of 1,3,5‐trimethylbenzene (TMB) is shown to be isomorphic to the group S3[C3] = C3 S3 of order 162, where denotes the wreath product of groups, and C3 is the cyclic group of order three and S3 is the symmetric group of order six on three letters. This group has 22 conjugacy classes and irreducible representations. The character table of the full nonrigid TMB is then derived for the first time. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Using nonrigid group theory, it is shown that the full nonrigid (f‐NRG) group of tetramethylsilane is isomorphic to the group C3 S4 of order 1944, where “” denotes the wreath product of groups, and C3 and S4 are the cyclic group of order three and the symmetric group of order four, respectively. This group has 51 conjugacy classes and irreducible representations. Then the character table of tetramethylsilane is derived for the first time. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Notation: Functions which are to be elements of a group will be indicated by small letters, e.g., z′ = f(z), z′ = d(z) etc., the identity-function being denoted by z′ = e(z) ; z and z′ will be used for the variables. Small letters may also be used for constants (real or complex), with no fear of ambiguity. Groups will be abbreviated: C n (cyclic group of order n ); D n (dihedral group of order 2n ); S n (symmetric group of order n! ); A n (alternating group of order .

Let D = FlxF2x- xF" be a direct product of n free groups Ft, F2, , Fn, a an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed and T an infinite cyclic group.Let D x T be the semidirect product of D and T with respect to a.We prove that the Whitehead group of D x T and the projective class group of the integral group ring Z(D x T) are trivial.The second result implies that the projective class group of the integral group ring over the fundamental group of a surface is trivial.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 K0Z(G).Let a be an automorphism of G and let F be an infinite cyclic group.Then we denote by G x x F the semidirect product of G and F with respect to a.Let M be a connected 2-dimensional manifold and trx(M) the fundamental group of M. If M is open, then ttx(M) is a free group so that K0Z(ttx(M)) is trivial by a theorem of Bass (cf.[I]).Next, if Misasphereor a projective plane, then trx(M)=0 or T2 (cyclic group of order 2) and so K0Z(ttx(Mj) =0 (cf.[7, p. 419]).Now, if Mis closed and is not a sphere or projective plane, then Farrell-Hsiang[4] have shown that trx(M) is just the semidirect product F xxT, where F is a free group.The purpose of this paper is to show that K0Z(F xx T)=0 and so the projective class group of the integral group ring over the fundamental group of a surface is always trivial.In fact, we prove:

Describing the group of units of a group ring is a classical problem. Let [Formula: see text] be a rational prime number. We set [Formula: see text] a primitive root of unity of order [Formula: see text], [Formula: see text] the ring of [Formula: see text]-cyclotomic integers, [Formula: see text] a finite abelian [Formula: see text]-group and [Formula: see text] the group of the units [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the augmentation map. We will prove that all the elements of the group [Formula: see text] arise from the units of the group ring [Formula: see text], where [Formula: see text] is the cyclic group of order [Formula: see text]. As an application, we describe explicitly the group of units of the group ring [Formula: see text] when [Formula: see text] is an elementary abelian [Formula: see text]-group and [Formula: see text] is a regular prime number.

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 V(RG) denote the normalized unit group of the group ring RG of a group G over a ring R. The concept of G-nilpotent unit in a commutative group ring has been defined in (Danchev, 2012). In this study, some necessary and sufficient conditions for a normalized unit group in a commutative group ring of a direct product group G×H to consist only of G×H-nilpotent units have been given and especially some results which are related to groups G×C_3 and G×C_4 have been introduced where C_3 and C_4 are cyclic groups of orders 3 and 4 respectively. In this context, we can say that the paper extends the results in (Danchev, 2012). At the end, an open problem is served as a future work.

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.

It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring \(F_{5}S_{4}\). The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F 3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F 2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F 2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.KeywordsGroup codeAbelian group codeSemisimplicity

Semi-clean group rings