Profinite groups of finite cohomological dimension

Let 1→N→G→G/N→1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n:=cd_p(G)<∞ and if the order of $\text{H}{}^{\mathrm{k}}\text{(N,}\text{F}{}_{\mathrm{p}}\text{)}$ is finite for k:=cd_p(N), the virtual cohomological p-dimension of G/N equals n?k. To cite this article: T. Weigel, P. Zalesskii, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Profinite groups associated with weakly primitive substitutions

A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword that is a factor of all its infinite factors, i.e., one that lies in a -class with only finite words strictly -above it. Such a -class is regular, and therefore it has an associated profinite group, namely, any of its maximal subgroups. One way to produce such -classes is to iterate finite weakly primitive substitutions. This paper is a contribution to the computation of the profinite group associated with the -class that is generated by the infinite iteration of a finite weakly primitive substitution. The main result implies that the group is a free profinite group provided the substitution induced on the free group on the letters that appear in the images of all of its sufficiently long iterates is invertible. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 13–48, 2005.     

Finite groups with 2-closed or 2′-closed centralizers of involutions

Finite nonsolvable groups are described whose involution centralizers are 2-closed or 2-closed, whereas the Sylow p-subgroups for p>2 are cyclic.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 437–446, October, 1971.In conclusion, the author expresses his gratitude to A. I. Starostin for posing the problem and for scientific guidance.     

Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct productA of quasi-cyclicp-groups with finitely many factors for each primep such that each of its elements does not commute elementwise with only finitely many Sylow subgroups ofA. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved. Translated fromMatematicheskie Zametki, Vol. 62, No 4, pp. 577–587, October, 1997. Translated by A. I. Shtern     

Infinite groups with Abelian centralizers of involutions

The article contains two characterizations of projective linear groups PGL₂(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second, in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which are recognizable by the set of their element orders. In memory of Viktor A. Gorbunov Supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 74–86, January–February, 2000.     

On abelian groups with commuting monomorphisms

We find necessary and sufficient conditions under which an abelian group with commuting monomorphisms has the endomorphism ring normal. We describe divisible groups with left- or righ-tinvariant monomorphisms and prove that a reduced torsion-free group with left- or right-invariant monomorphisms has the endomorphism ring normal. Necessary and sufficient conditions are indicated for the left and right invariance of monomorphisms from some classes of splitting groups.     

Computing the optimal commuting matrix pairs

A matrix can be modified by an additive perturbation so that it commutes with any given matrix. In this paper, we discuss several algorithms for computing the smallest perturbation in the Frobenius norm for a given matrix pair. The algorithms have applications in 2-D direction-of-arrival finding in array signal processing. The work of first author was supported in part by NSF grant CCR-9308399. The work of the second author was supported in part by China State Major Key Project for Basic Researches.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.