Learning with errors over group rings constructed by semi-direct product

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.

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

A theorem of a rather general nature is proved, which gives a positive solution to the restricted Burnside problem for a variety of groups with operators whose identities are obtained by “operator diluting” (in some precise sense) ordinary identities defining a variety of groups for which this problem has a positive solution. Namely, let be a finite group, a family of -operator identities, and a family of (ordinary) group identities obtained from by replacing all operators by 1. Suppose that the associated Lie ring of a free group in the variety defined by satisfies a system of multilinear identities that defines a locally nilpotent variety of Lie rings with a function bounding the nilpotency class of a -generator Lie ring in this variety. It is proved that if, for a -generator -group , the semidirect product is nilpotent, then the nilpotency class of is at most . A strong condition that be nilpotent is automatically satisfied if both and are finite -groups. Instead of the condition on the identities of the associated Lie ring, an analogous condition on the identities could be required, but such a condition would be stronger. An example at the end of the paper shows that the word multilinear in this condition is essential. It is not yet clear whether the condition that be finite is essential, and whether one can choose a function from the conclusion to be independent of . Earlier, in [1], a similar theorem on nilpotency in varieties of groups with operators was proved by the author. The author's results on groups with splitting automorphisms of prime order (see [2], [3]) are prototypes for both papers on operator groups. Bibliography: 18 titles.

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

Describing the group of units U(ZG) of the integral group ring ZG, for a finite group G, is a classical and open problem. In this paper, it is shown that U(Z[G × Cp]) = M ⋊ U(ZG), a semi-direct product where M is a certain subgroup of U(Z[ζ]G) and p prime. For p = 2, this structure theorem is applied to give precise descriptions of U(ZG) for a non-abelian group G of order 32, G = C10, and G = C8 × C2.

Augmentation quotients of integral group rings II

Let G be a group, ZG the integral group ring of G and Δ(G) its augmentation ideal. M. Khambadkone [2] studies the quotient group Δ(G) Δ(H)/Δ 2 (G) Δ(H) when is a normal subgroup of G and gives the result that if G is the semidirect product H |>K of a finitely generated normal subgroup H by a subgroup K, then Δ(G) Δ(H)/Δ 2 (G) Δ(H) ≅ K|K'⊗ HIH' ⊕

A note on set-theoretic solutions of the Yang–Baxter equation

The independence assumption for a series or parallel system when component lifetimes are exponential : John P. Klein and M. L. Moeschberger. IEEE Trans. Reliab.R-35 (3), 330 (1986)

In this paper we investigate the matrix ring structure of skew group rings in some particular cases. We prove that if G is a finite group acting on a finite field K with abelian kernel N, then the skew group ring R = K ∗ θ G is isomorphic to M m ( Z ( R ) ) , where m = [ G : N ] and Z ( R ) is the center of R. We give also the matrix ring structure of K ∗ θ G when K has a prime characteristic p and the kernel is a p-group. At the end, a result on the matrix ring structure of K ∗ θ G when G is a semi-direct product is also presented.

On Fox and augmentation quotients of semidirect products

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

Torsion units 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.

The structure of the group of automorphisms of the integer group ring of the group A4 is studied in terms of a semidirect product. We show that the Zassenhaus conjecture on the structure of automorphisms of integer group rings of finite groups for the group Aut ℤA4 holds.