# Rendiconti del Seminario Matematico della Università di Padova

## Forthcoming Articles

(Last update: May 25, 2018)

The following articles will soon be published in this journal. Their publication status can be :

• online first – final version available (without page numbers nor DOI);
• accepted – manuscript available.

To view the abstract, or download the full paper in PDF[1], click on the article's title.

## Online first

•

Abstract: See the paper itself.

## Accepted Papers

• ### Generating sets of Galois equivariant Krasner analytic functions

Department of Mathematics
University of Bucharest
RO-010014 Bucharest
Romania

Simion Stoilow Institute of Mathematics of the Romanian Academy
P.O. Box 1-764
RO-014700 Bucharest
Romania

Simion Stoilow Institute of Mathematics of the Romanian Academy
P.O. Box 1-764
RO-014700 Bucharest
Romania

Abstract: Given a prime number $p$ and $x$ an element of the Tate field $\mathbb{C}_p$, the main goal of the present paper is to provide an explicit generating set, which is given by the trace function of $x$ and all its derivatives, for the $\mathbb{C}_p$-Banach algebra of the Galois equivariant Krasner analytic functions defined on the complement in $\mathbb{P}^1(\mathbb{C}_p)$ of the orbit of $x$ with values in $\mathbb{C}_p$.

received 26.09.2017, accepted 13.03.2018 (14 pages)
• ### Nadler mappings in cone $b$-metric spaces over Banach algebras

Mathematics
Yildiz Technical University
34210, Istanbul
Turkey

Abstract: In this study we first define the concept of Nadler type contraction in the setting of $H$-cone $b$-metric space with respect to cone $b$-metric spaces over Banach algebras. Next we prove the Banach contraction principle for such contractions by means of the notion of spectral radius and a solid cone in underlying Banach algebra. Finally we observe that the main result achieved in this work extends and generalizes the well known results associated with contractions of Nadler type.

received 05.02.2018, accepted 12.03.2018 (10 pages)
• ### Structure d algebre de Gerstenhaber-Voronov sur les formes differentielles non commutatives

Institut prÇparatoire aux Çtudes d ingÇnieurs de Nabeul
UniversitÇ de Carthage
Campus Universitaire
Merazka, 8000
Nabeul
Tunisie

hatem_iaaoui@yahoo.com

DÇpartement des MathÇmatiques
FacultÇ des sciences de Tunis
UniversitÇ Tunis El Manar

naoufelbattikh@yahoo.fr

Abstract: The algebra of noncommutative differential forms has been defined by A. Connes in }$\left[ 4\right] .$ Using this algebra, M. Karoubi has defined cyclic homology and Hochschild homology groups (see. $\left[ 13\right]$). These groups are related to the algebraic K-theory. The purpose of this paper is to provide the noncommutative differential forms algebra with the structure of Gerstenhaber-Voronov algebras.

received 13.06.2017, accepted 01.03.2018 (15 pages)
• ### On hypercentre-by-polycyclic-by-nilpotent groups

School of Mathematical Sciences
Queen Mary University of London
London E1 4NS
England

b.a.f.wehrfritz@qmul.ac.uk

Abstract: If $\{\gamma^{s+1}G\}$ and $\{\zeta_s(G)\}$ denote respectively the lower and upper central series of the group $G$, $s\ge 0$ an integer, and if $\gamma^{s+1}G/(\gamma^{s+1}G\cap\zeta_s(G))$ is polycyclic (resp. polycyclic-by-finite) for some $s$, then we prove that $G/\zeta_{2s}(G)$ is polycyclic (resp. polycyclic-by-finite). The corresponding result with polycyclic replaced by finite was proved in 2009 by G. A. Fern†ndez-Alcober and M. Morigi. We also present an alternative approach to the latter.

received 25.08.2017, accepted 23.02.2018 (10 pages)
• ### On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones

Departamento de Fisica
Av. Peter Henry Rolfs s/n
Campus Universitario
Viáosa, MG
Brasil, CEP: 36.570-900

daniel.franco@ufv.br

universidade Federal de Juiz de Flora
Departamento de Matematica
Campus Universitario
Bairro Martelos
Juiz de Flora, MG
Brasil, CEP: 36.036-900

magno_branco@yahoo.com.br

Abstract: We address in this paper the issue of singularities of ultrahyperfunctions. Following the Carmichael's approach for ultrahyperfunctions, we study the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In passing, a simple version of the celebrated edge of the wedge theorem for this setting is derived from the integral representation without using cohomology.

received 18.08.2017, accepted 09.02.2018 (17 pages)
• ### Twisted cyclic groups

Department of Mathematics and Statistics
Youngstown State University
One University Plaza
Youngstown, OH 44555
United States

Abstract: A finite group $G$ is said to be \textsl{twisted cyclic} if there exist $\phi \in \text{Aut}(G)$ and $x \in G$ such that $G=\{(x^i)\phi^j\ :\ i,j \in \mathbb{Z}\}$. In this note, we classify all groups satisfying this property and determine that, if a finite group $G$ is twisted cyclic, then $G$ is isomorphic to $\mathbb{Z}_{p^n}$, $\mathbb{Z}_p\times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$, $Q_8$, $\mathbb{Z}_{p^n}\times \mathbb{Z}_{p^n}$ or direct products of these groups for some prime $p$ and some $n\in \mathbb{Z}^{+}$.

received 03.01.2018, accepted 09.02.2018 (9 pages)
• ### On the values of unipotent characters in bad characteristic

IAZ - Lehrstuhl fÅr Algebra
UniversitÑt Stuttgart
Pfaffenwaldring 57
D-70569 Stuttgart
Germany

meinolf.geck@mathematik.uni-stuttgart.de

Abstract: Let $G(q)$ be a Chevalley group over a finite field $\F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach.

received 24/01/2018, accepted 06.02.2018 (20 pages)
• ### Finite groups with the pp-embedding property

Institute of Mathematics
University of Bialystok
K. Ciolkowskiego 1M
15-245 Bialystok
Poland

Abstract: A subset $X$ of a finite group $G$ is called g-independent if there is no proper subset $Y$ of $X$ such that $\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle.$ The group $G$ has the embedding property if every g-independent subset of $G$ can be embedded in a minimal generating set of $G$. If $X$ is a set of prime power order elements, then we say that $G$ has the pp-embedding property. In this note we classify all finite solvable groups with the pp-embedding property. Moreover we prove that this class is equal to the class of finite solvable groups with the embedding property.

received 26.10.2017, accepted 19.01.2018 (11 pages)
• ### Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Department of Applied Mathematics
Northwestern Polytechnical University
Xian
China

Abstract: See the article

received 18.03.2017, accepted 09.12.2017 (32 pages)
• ### On the generalized $\sigma$-Fitting subgroup of finite groups

Hu, Huang

School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou, 221116
P. R. China

Skiba

Department of Mathematics
Francisk Skorina Gomel State University
Gomel 246019
Belarus

Abstract: see the article

received 26.02.2017, accepted 29.11.2017 (14 pages)
• ### On the monotonicity of Hilbert functions

Department of Mathematics
Indian Institute of Technology Bombay
Powai
Mumbai 400 076
India

Abstract: In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ (with respect to $\mathfrak{m}$) is non-decreasing. Examples include
(1) Complete intersections $A = Q/(f,g)$ where $(Q,\mathfrak{n})$ is regular local of dimension three and $f \in \mathfrak{n}^2 \setminus \mathfrak{n}^3$.
(2) One dimensional Cohen-Macaulay quotients of a two dimensional Cohen-Macaulay local ring with pseudo-rational singularity.

received 16.02.2017, accepted 09.11.2017 (7 pages)
• ### Zeros of irreducible characters of metabelian $p$-groups

London SW11 6AA
UK

Abstract: We show that if $\chi$ is an irreducible complex character of a metabelian $p$ - group $P,$ where $p$ is an odd prime, and if $x\in P$ satisfies $\chi(x)\neq 0$, then the order of $x$ divides $|P|/\chi(1)^2.$

received 21.06.2017, accepted 09.11.2017 (9 pages)
• ### Cohen-Macaulayness and sequentially Cohen-Macaulayness of monomial ideals

Department of Mathematics
Tabriz
Iran

Department of Mathematics
Razi University
Kermanshah
Iran

Abstract: In this paper, we give a characterization for Cohen--Macaulay rings $R/I$ where $I\subset R=K[y_1, \dots, y_n]$ is a monomial ideal which satisfies $\rm{bigsize}\, I=\rm{size}\, I$. Next, we let $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a polynomial ring and $I\subset S$ a monomial ideal. We study the sequentially Cohen-Macaulayness of $S/I$ with respect to $Q=(y_1,\ldots,y_n)$. Moreover, if $I\subset R$ is a monomial ideal such that the associated prime ideals of $I$ are in pairwise disjoint sets of variables, a classification of $R/I$ to be sequentially Cohen-Macaulay is given. Finally, we compute $\rm{grade}\,(Q, M)$ where $M$ is a sequentially Cohen-Macaulay $S$-module with respect to $Q$.

received 02.04.2017, accepted 23.10.2017 (14 pages)
• ### On generalized $\Pi$-property of subgroups of finite groups

School of Mathematics
Jilin University
No. 2699 Qianjin Street
Changchun
China

Abstract: In this note, we extend the concept of $\Pi$-property of subgroups of finite groups and generalize some recent results. In particular, we generalize the main results of Li et al. [8] and Miao et al. [9].

received 28.08.2017, accepted 23.10.2017 (17 pages)
• ### On the uniqueness of the factorization of power digraphs modulo n

Department of Mathematics
North-Eastern Hill University
Shillong-793022
India

Department of Basic Sciences & Social Sciences
North-Eastern Hill University
Shillong-793022
India

Abstract: For each pair of integers $n=\prod_{i=1}^{r}p_i^{e_i}$ and $k \geq 2$, a digraph $G(n,k)$ is one with vertex set $\{0,1,\ldots ,n-1\}$ and for which there exists a directed edge from $x$ to $y$ if $x^k \equiv y \pmod n$. Using the Chinese Remainder Theorem, the digraph $G(n,k)$ can be written as a direct product of digraphs $G(p_i^{e_i},k)$ for all $i$ such that $1 \leq i \leq r$. A fundamental constituent $G_{P}^{*}(n,k)$, where $P \subseteq Q=\{p_1,p_2,\ldots,p_r\}$, is a subdigraph of $G(n,k)$ induced on the set of vertices which are multiples of $\prod_{{p_i} \in P}p_i$ and are relatively prime to all primes $p_j \in Q \smallsetminus P$. In this paper, we investigate the uniqueness of the factorization of trees attached to cycle vertices of the type $0$, $1$, and $(1,0)$, and in general, the uniqueness of $G(n,k)$. Moreover, we provide a necessary and sufficient condition for the isomorphism of the fundamental constituents $G_{P}^{*}(n,k_1)$ and $G_{P}^{*}(n,k_2)$ of $G(n,k_1)$ and $G(n,k_2)$ respectively for $k_1 \neq k_2$.

received 01.06.2017, accepted 20.10.2017 (28 pages)
• ### Semisimple groups that are quasi-split over a tamely ramified extension

Université Claude Bernard Lyon 1
CNRS UMR 5208
Institut Camille Jordan
43 Blvd. du 11 novembre 1918
F-69622 Villeurbane cedex
France

Abstract: Let $K$ be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple $K$-group $G$ is quasi-split if and only if it quasi-splits after a finite tamely ramified extension of $K$.

received 07.07.2017, accepted 16.10.2017 (9 pages)
• ### Complex manifolds as families of homotopy algebras

Institut de Mathématiques de Toulouse
UMR5219, Université de Toulouse
CNRS, UPS IMT
F-31062 Toulouse Cedex 9
France

Abstract: We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex structure on a smooth manifold corresponds in this way to a family of algebras indexed by the points of the manifold.

received 24.03.2017, accepted 25.09.2017 (38 pages)
• ### A brief journey through extensions of rational groups

University of Stralsund
Zur Schwedenschanze 15
18435 Stralsund
Germany

Abstract: Let $A$ and $B$ be rational groups, i.e. torsion-free groups of rank-1 and thus subgroups of the rational numbers. This paper gives a short overview of the structure of $\text{Ext}\,(A,B)$ especially considering some interesting classes of torsion-free pairs.

received 06.03.2017, accepted 20.09.2017 (5 pages)
• ### Algebraic cycles on a very special EPW sextic

Institut de Recherche Mathématique Avancée
CNRS _ Université de Strasbourg
7 Rue René Descartes
67084 Strasbourg CEDEX
France

Abstract: Motivated by the Beauville--Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten--Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperk\"ahler fourfolds.

received 22.06.2016, accepted 28.08.2017 (27 pages)
• ### Algèbres de distributions et ${\cal D}$-modules arithmétiques

IRMA
Université de Strasbourg
7, rue René Descartes
67084 Strasbourg cedex
France

IRMAR
Université de Rennes 1
Campus de Beaulieu
35042 Rennes cedex
France

Abstract: Let $p$ be a prime number, $V$ a complete discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level $m$, generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension $K$ of ${\bf Q}_p$ . We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$, and prove a coherence result when the ramification index of $K$ is $received 13.10.2015, accepted 07.07.2017 (66 pages) download: manuscript (778K) • ### Countable recognizability and residual properties of groups Dipartimento di Matematica e Applicazioni Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia, Napoli Italy Abstract: A class of groups {\mgoth X} is said to be countably recognizable if a group belongs to~{\mgoth X} whenever all its countable subgroups lie in {\mgoth X}. It is proved here that the class of groups whose subgroups are closed in the profinite topology is countably recognizable. Moreover, countably detectable properties of the finite residual of a group are studied. received 17.01.2017, accepted 29.05.2017 (11 pages) download: manuscript (306K) • ### A result on a singular Cauchy problem with a radial point revisited using microdifferential calculus Department of Mathematics Edificio Mario Laserna Cra 1 Este No 19A 40 Bogota Colombia Universidad de los Andes Colombia Abstract: We study the ramified Cauchy problem for a linear PDE with a radial point using the theory of microdifferential operators. received 23.09.2016, accepted 11.05.2017 (20 pages) download: manuscript (447K) • ### Mostow s fibration for canonical embeddings of compact homogeneous$CR$manifolds Dipartimento di Matematica e Fisica III Università di Roma Largo San Leonardo Murialdo, 1 00146 Roma Italy Dipartimento di Matematica II Università di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Roma Italy Abstract: We define a class of compact homogeneous$CR$manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy-Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings. received 17.12.2016, accepted 09.05.2017 (30 pages) download: manuscript (278K) • ### Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries RUDN University S. M. Nikol’skii Institute 6 Miklukho-Maklay St Moscow, 117198 Russia Abstract: We introduce a class of Morrey-type spaces$M^\lambda_{p,q}$, which includes classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem. This theorem is then applied to obtaining the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operator. received 05.10.2016, accepted 26.04.2017 (20 pages) download: manuscript (173K) • ### Motifs et adjoints Institut de Mathématiques de Jussieu - PRG Case 247 4 place Jussieu 75252 Paris Cedex 05 France Abstract: We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrasted with the more classical situation where one deals with a finite type morphism of schemes. Among various applications, one is a construction of a Tate-\v Safarevi\v c motive" attached to an abelian variety over a function field. We also deduce a possible approach to \text{Bloch's} conjecture on surfaces, by reduction to curves. received 18.11.2015, accepted 22.03.2017 (44 pages) download: manuscript (702K) • ### Definable categories and$\mathbb{T}$-motives Dipartimento di Matematica "F. Enriques" Università degli Studi di Milano Via C. Saldini, 50 I-20133 Milano Italy School of Mathematics University of Manchester Oxford Road Manchester M13 9PL UK Abstract: Making use of Freyd's free abelian category on a preadditive category we show that if$T:D\rightarrow \mathcal{A}$is a representation of a quiver$D$in an abelian category$\mathcal{A}$then there is an abelian category$\mathcal{A} (T)$, a faithful exact functor$F_T: \mathcal{A} (T) \to \mathcal{A}$and an induced representation$\tilde T: D \to \mathcal{A} (T)$such that$F_T\tilde T= T$universally. We then can show that$\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories. received 03.10.2016, accepted 15.03.2017 (18 pages) download: manuscript (395K) • ### Nunke’s problem for abelian$p$-groups of uncountable length Department of Mathematics Whitman College Walla Walla, WA 99362 USA Abstract: \textit{Nunke's Problem} asks when the torsion product of two abelian$p$-groups is a direct sum of countable reduced groups. In previous work the author gave a complete answer to this question when the groups involved have countable length. In this paper a complete answer is given in the case of groups of uncountable length, at least in any set-theoretic universe in which$2^{\aleph_1}=\aleph_2$. received 18.01.2017, accepted 03.03.2017 (18 pages) download: manuscript (292K) • ### The isoperimetric problem in Grushin space$\mathbb{R}^{h+1}$with density$|x|^p$School of Science Nanjing University of Science and Technology Nanjing 210094 P. R. China Abstract: In this paper we study the isoperimetric problem in a class of$x$-spherically symmetric sets in the Grushin space$\mathbb{R}^{h+1}$with density$|x|^p$,$p>-h+1$. First we prove the existence of weighted isoperimetric sets. Then we deduce that, up to a vertical translation, a dilation and a negligible set, the weighted isoperimetric set is only of the form$\big\{(x,y)\in \mathbb{R}^{h+1}:|y|<\int_{\arcsin|x|}^{\frac{\pi}{2}}\sin^{\alpha+1}(t)dt,\hspace{2mm}|x|<1\big\}$. received 31.01.2017, accepted 24.02.2017 (17 pages) download: manuscript (423K) • ### Cyclic subgroup commutativity degrees of finite groups Faculty of Mathematics Al. I. Cuza University Iasi Romania Abstract: In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group$G$. This quantity measures the probability of two random cyclic subgroups of$G$commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented. received 13.12.2016, accepted 21.02.2017 (15 pages) download: manuscript (311K) • ### Weak local-global compatibility and ordinary representations Instytut Matematyczny PAN Ul. Sniadeckich 8 00-656 Warsawa Poland Abstract: We introduce a general formalism with minimal requirements under which we are able to prove the pro-modular Fontaine-Mazur conjecture. We verify it in the ordinary case using the recent construction of Breuil and Herzig. received 04.07.2016, accepted 13.02.2017 (22 pages) download: manuscript (436K) • ### Some sufficient conditions for p-nilpotence of a finite group School of Mathematics and Statistics Jiangsu Normal University Xuzhou, 221116 China lcw2000@126.com Abstract: In this paper, we give some new characterizations of finite$p$-nilpotent groups by using the notion of$\mathcal{H}C$-subgroups and extend several recent results. received 28.09.2016, accepted 17.01.2017 (8 pages) download: manuscript (98K) • ### On an open problem about$\pi$-quasi-$\mathcal{F}$-groups Department of Mathematics University of Science and Technology of China Hefei, 230026 P.R. China Department of Mathematics University of Science and Technology of China Hefei, 230026 P.R. China School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou, 22 1116 P.R. China Abstract: Let$\mathfrak{F}$be a class of finite groups,$p$a prime and$\pi$a set of some primes. A finite group$G$is called a$p$-quasi-$\mathfrak{F}$-group (respectively, by$\pi$-quasi-$\mathfrak{F}$-group) provided that for every$\mathfrak{F}$-eccentric$G$-chief factor$H/K$of order divisible by$p$(respectively, by at least one prime in$\pi$), the automorphisms of$H/K$induced by all elements of$G$are inner. In this paper, we obtain the characterizations of$p$-quasi-$\mathfrak{F}$-groups and$\pi$-quasi-$\mathfrak{F}$-groups, which give a positive answer to an open problem in the book [3]. received 29.07.2016, accepted 04.11.2016 (8 pages) download: manuscript (336K) • ### On$W$-$S$-permutable subgroups of finite groups Department of Mathematics Shanghai University Shanghai 200444 P. R. China Abstract: A subgroup$H$of a finite group$G$is said to be$W$-$S$-permutable in$G$if there is a subgroup$K$of$G$such that$G=HK$and$H\cap K$is a nearly$S$-permutable subgroup of$G$. In this article, we analyse the structure of a finite group$G$by using the properties of$W$-$S$-permutable subgroups and obtain some new characterizations of finite$p$-nilpotent groups and finite supersolvable groups. Some known results are generalized. received 20.04.2016, accepted 06.10.2016 (12 pages) download: manuscript (355K) • ### On$H_\sigma$-permutably embedded subgroups of finite groups Guo Zhang Department of Mathematics University of Science and Technology of China Hefei 230026 P. R. China Skiba Sinitsa Department of Mathematics Francisk Skorina Gomel State University Gomel 246019 Belarus Abstract: Let$G$be a finite group. Let$\sigma =\{\sigma_{i} | i\in I\}$be a partition of the set of all primes$\Bbb{P}$and$n$an integer. We write$\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$,$\sigma (G) =\sigma (|G|)$. A set$ {\cal H}$of subgroups of$G$is said to be a complete Hall$\sigma $-set of$G$if every member of${\cal H}\setminus \{1\}$is a Hall$\sigma _{i}$-subgroup of$G$for some$\sigma _{i}$and${\cal H}$contains exact one Hall$\sigma _{i}$-subgroup of$G$for every$\sigma _{i}\in \sigma (G)$. A subgroup$A$of$G$is called: (i) a$\sigma$-Hall subgroup of$G$if$\sigma (A) \cap \sigma (|G:A|)=\emptyset$; (ii)${\sigma}$-permutable in$G$if$G$possesses a complete Hall$\sigma$-set${\cal H}$such that$AH^{x}=H^{x}A$for all$H\in {\cal H}$and all$x\in G$. We say that a subgroup$A$of$G$is$H_{\sigma}$-permutably embedded in$G$if$A$is a${\sigma}$-Hall subgroup of some${\sigma}$-permutable subgroup of$G$. We study finite groups$G$having an$H_{\sigma}$-permutably embedded subgroup of order$|A|$for each subgroup$A$of$G\$. Some known results are generalized.

received 07.06.2016, accepted 18.08.2016 (13 pages)