Constructible Matrix Groups

We prove that the additive group of a ring K is constructible if the group GL ₂ (K) is constructible. It is stated that under one extra condition on K, the constructibility of GL ₂ (K) implies that K is constructible as a module over its subring L generated by all invertible elements of the ring L; this is true, in particular, if K coincides with L, for instance, if K is a field or a group ring of an Abelian group with the specified property. We construct an example of a commutative associative ring K with 1 such that its multiplicative group K* is constructible but its additive group is not. It is shown that for a constructible group G represented by matrices over a field, the factors w.r.t. members of the upper central series are also constructible. It is proved that a free product of constructible groups is again constructible, and conditions are specified under which relevant statements hold of free products with amalgamated subgroup; this is true, in particular, for the case where an amalgamated subgroup is finite. Also we give an example of a constructible group GL ₂ (K) with a non-constructible ring GL. Similar results are valid for the case where the group SL ₂ (K) is treated in place of GL ₂ (K) .     

Enumerating Groups Acting Regularly on a d-Dimensional Cube

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G _p  × B a finite group, where G _p is a p-group and B is a p′-group. Denote by K ^λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z ²(G, K*). In this article, we give necessary and sufficient conditions for K ^λ G to be of OTP representation type, in the sense that every indecomposable K ^λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K ^λ G _p -module V and an irreducible K ^λ B-module W.     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

Finite Groups Whose Minimal Subgroups are c-Supplemented

Let G be a finite group. A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _G , where H _G  = Core _G (H) is the largest normal subgroup of G contained in H. In this article, we investigate the structure of a finite group G under the assumption that subgroups of prime order are c-supplemented in G. Moreover, we analyze the structure of a group G when the minimal subgroups of the generalized Fitting subgroup F?(G) of G are c-supplemented in G through the theory of formations.     

Centralizers of Involutions in Locally Finite Groups

The present article deals with locally finite groups G having an involution φ such that C _G (φ) is an SF-group. It is shown that G possesses a normal subgroup B which is a central product of finitely many groups isomorphic to PSL(2, K _i ) or SL(2, K _i ) for some infinite locally finite fields K _i of odd characteristic, such that [G, φ]′/B and G/[G, φ] are both SF-groups.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GL _n(K) (general), SL _n(K) (special), and UT _n(K) (unitriangular) over K. It is proved that for n 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UT _n(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.     

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group H_p. It is proved that the subgroup E of the automorphism group Aut(H_p) of H_p generated by those transvections is normal in it, and that Aut(H_p) can be written as the product of E and some abelian subgroup K. The center of Aut(H_p) is also determined.     

Normal Restriction in Finite Groups

A subgroup H of a finite group G is called an NR ? subgroup (Normal Restriction) if whenever K ? H, then K ^G ∩ H = K, where K ^G is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups.     

On Nearly S-Permutable Subgroups of Finite Groups

We say that a subgroup H of a finite group G is nearly S-permutable in G if for every prime p such that (p, |H|) = 1 and for every subgroup K of G containing H the normalizer N _K (H) contains some Sylow p-subgroup of K. We study the structure of G under the assumption that some subgroups of G are nearly S-permutable in G.     

A Characteristic Property of Finite p -Nilpotent Groups

Abstract

An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\K. It is proved that a finite group G is p-nilpotent if and only if every element in G _p  \Φ(G _p ) is Q-central. We will adapt Doerk and Hawkes [Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.     

Groups with bounded representation degree

Let Gbe a group and let K be a field of characteristic p> 0. If all irreducible representations of the group algebra K[G] have finite degree < n, then we show that G has a subgroup A with |G:A| bounded by a function of nand such that all the irreducible representations of K[A] have degree 1.     

A Decomposition Theorem for CK1 of Central Simple Algebras

Let A be a central simple algebra over its center F. Define CK₁ A = Coker(K₁ F → K₁ A). We prove that if A and B are F-central simple algebras of coprime degrees, then CK₁(A? _F B) = CK₁ A × CK₁ B.     

The Selmer Groups of Elliptic Curves

Let E/K be an elliptic curve with K-rational p-torsion points.The p-Selmer group of E is described by the image of a map λk and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.     

A Singular Analogue of Gersten's Conjecture and Applications to K-theoretic Adèles

The first part of this article introduces an analogue, for one-dimensional, singular, complete local rings, of Gersten's injectivity conjecture for discrete valuation rings. Our first main result is the verification of this conjecture when the ring is reduced and contains ?, using methods from cyclic/Hochschild homology and Artin–Rees results due to A. Krishna.

The second part of the article describes the relationship between adelic resolutions of K-theory sheaves on a one-dimensional scheme and properties of K-theory such as localization and descent. In particular, we construct a new resolution of Nisnevich sheafified K-theory, conditionally upon the aforementioned conjecture.     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.

     

Local c*-Supplementation of Some Subgroups in Finite Groups

A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation.