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This paper has two souls. On one side, it is a survey on (infinite) groups in which certain systems of subgroups are complemented (like for instance the abelian subgroups). On another side, it provides generalizations and new, easier proofs of some (un)known results in this area.

In this paper we characterize, in terms of their conjugacy classes, linear groups G such that $$G/\zeta _k(G)$$ G / ζ k ( G ) belongs to a certain group class $$\mathfrak {X}$$ X for several natural choices of $$\mathfrak {X}$$ X . Moreover, a description is given of linear groups with restrictions on layers.

A classical theorem of Reinhold Baer shows that any group which is finite over its k-th centre has a finite (k + 1)-th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index $|G:\zeta_{2k}(G)|$ is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by Yuri${\text{\u{\i}}}$ Merzljakov that Baer’s theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions. Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups.

The aim of this paper is to investigate groups whose proper subgroups are linear. Although there exist locally nilpotent non-linear groups in which every proper subgroup is linear, we prove that, for a large group class containing in particular all hypercentral groups, the linearity of the proper subgroups implies that of the whole group. Moreover, we study groups whose proper subgroups are linear of characteristic 0, showing in particular that these groups are themselves linear of characteristic 0 provided that they are either locally nilpotent or locally finite.

Let Ω be an open convex set in Rm with finite width, and with boundary ∂Ω. Let vΩ be the torsion function for Ω, i.e., the solution of −Δv=1,v|∂Ω=0. An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that ∥vΩ∥L1(Ω)λ(Ω)≥π2/24, and that this bound is sharp.

This paper provides an account of results and methods from the theory of infinite groups admitting only finitely many normalizers of subgroups with a given property. Some new statements on this subject are also proved.

Countably recognizable group classes were introduced by Reinhold Baer and provide a very ingenious way to study large groups through the properties of their countable subgroups. This is the reason we have chosen the countable recognizability to start this series of survey papers on infinite group theory.

In this paper we define analogues of the Wielandt subgroup of a group. We say that a subgroup H of a group G is f-subnormal in G if there is a finite chain of subgroups $$ H=H_0\le H_1\le \cdots \le H_n=G $$ such that either $$|H_{i+1}: H_i|$$ is finite or $$H_i$$ is normal in $$H_{i+1}$$, for $$0\le i\le n-1$$. We study in a group G the connection between the subgroups $$\overline{\omega }(G)$$ and $$\overline{\omega }_i(G)$$ which are, respectively, the sets of elements of G normalizing all f-subnormal subgroups of G and those normalizing all infinite f-subnormal subgroups of G. In particular we show that $$\overline{\omega }_i(G)/\overline{\omega }(G)$$ is always Dedekind and often abelian.

Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$. Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then $\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1, Comment: One missing case (n=4) has been included in the appendix. Typos are corrected, Journal of Combinatorial Theory, Series A, 2018

A group G is said to be cohopfian if it is neither trivial nor isomorphic to any of its proper subgroups, and this property is equivalent to the existence of a suitable group class X such that G is minimal non-X. If X is any group class, the subclass X0 consisting of all groups that are isomorphic to proper subgroups of locally graded minimal non-X groups is often much smaller than X. Similarly, if Xprop is the class of all groups isomorphic to proper subgroups of X-groups, the class X of all locally graded minimal non-Xprop groups may contain many groups which are not in X. This paper investigates the relation between the classes X, X0 and X.

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

We study thermal insulating of a bounded body $\Omega\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $\Omega$ and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\varepsilon>0$, and whose conductivity is also proportional to $\varepsilon$. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the $\Gamma$-limit of the energy functional $F_\varepsilon$ and prove that this is a functional $F$ whose minimizers still satisfy an elliptic PDEs system with a non uniform Robin boundary condition depending on the distribution of insulating layer around $\Omega$. In a second step we study the maximization of heat content (which measures the goodness of the insulation) among all the possible distributions of insulating material with fixed mass, and prove an optimal upper bound in terms of geometric properties. Eventually we prove a conjecture which states that the ball surrounded by a uniform distribution of insulating material maximizes the heat content.

Let $${\mathfrak {X}}$$ X be a group class. A group G is an opponent of $${\mathfrak {X}}$$ X if it is not an $${\mathfrak {X}}$$ X -group, but all its proper subgroups belong to $${\mathfrak {X}}$$ X . Of course, every opponent of $${\mathfrak {X}}$$ X is a cohopfian group and the aim of this paper is to describe the smallest group class containing $${\mathfrak {X}}$$ X and admitting no such a kind of cohopfian groups.

Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the equivalence issue of maximum $d$-codes of Hermitian matrices. More precisely, in the space $\mathrm{H}_n(q^2)$ of Hermitian matrices over $\mathbb{F}_{q^2}$ we have two possible equivalence: the classical one coming from the maps that preserve the rank in $\mathbb{F}_{q^2}^{n\times n}$, and the one that comes from restricting to those maps preserving both the rank and the space $\mathrm{H}_n(q^2)$. We prove that when $d, Comment: Accepted for publication in Journal of Algebraic Combinatorics

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n\ge 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultanously the surface area, the volume and the boundary momentum of convex sets. As a by product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue, Comment: 1 figure

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta (Combin Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.

We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on L 1 comparison) between solutions to an elliptic partial differential equation on a smooth bounded set Ω with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as Ω. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.

A group is called metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite locally graded group whose proper subgroups are metahamiltonian is likewise metahamiltonian, and the aim of this paper is to describe the structure of locally graded groups whose proper subgroups contain a metahamiltonian subgroup of finite index.

If 𝔛 {\mathfrak{X}} is a class of groups, define a sequence of group classes 𝔛 1 , 𝔛 2 , … , 𝔛 k , … {\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots,\mathfrak{X}_{k},\ldots} by putting 𝔛 1 = 𝔛 {\mathfrak{X}_{1}=\mathfrak{X}} and choosing 𝔛 k + 1 {\mathfrak{X}_{k+1}} as the class of all groups whose non-permutable subgroups belong to 𝔛 k {\mathfrak{X}_{k}} . In particular, if 𝔄 {\mathfrak{A}} is the class of abelian groups, 𝔄 2 {\mathfrak{A}_{2}} is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of 𝔛 k {\mathfrak{X}_{k}} -groups, with special emphasis on the case 𝔛 = 𝔄 {\mathfrak{X}=\mathfrak{A}} . Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to 𝔄 k {\mathfrak{A}_{k}} for some positive integer k.

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.

In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set K, which represents a conductor of constant temperature, say 1, is thermally insulated by surrounding it with a layer of thermal insulator, the open set $$\Omega {\setminus } K$$ Ω \ K with $$K\subset \bar{\Omega }$$ K ⊂ Ω ¯ . The heat dispersion is then obtained as $$\begin{aligned} \inf \left\{ \int _{\Omega }|\nabla \varphi |^{2}dx +\beta \int _{\partial ^{*}\Omega }\varphi ^{2}d\mathcal H^{n-1} ,\;\varphi \in H^{1}(\mathbb R^{n}), \, \varphi \ge 1\text { in } K\right\} , \end{aligned}$$ inf ∫ Ω | ∇ φ | 2 d x + β ∫ ∂ ∗ Ω φ 2 d H n - 1 , φ ∈ H 1 ( R n ) , φ ≥ 1 in K , for some positive constant $$\beta $$ β .We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set K vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.

In this survey we consider the classical overdetermined problem which was studied by Serrin in 1971. The original proof relies on Alexandrov's moving plane method, maximum principles, and a refinement of Hopf's boundary point Lemma. Since then other approaches to the same problem have been devised. Among them we consider the one due to Weinberger which strikes for the elementary arguments used and became very popular. Then we discuss also a duality approach involving harmonic functions, a shape derivative approach and a purely integral approach, all of them not relying on maximum principle. For each one we consider pros and cons as well as some generalizations.

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class is F-characteristic if any group containing an -subg...

In this paper, we present a new family of maximum rank-distance (MRD) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$ . In particular, when $d=2n$ , we can show that the corresponding semifield is exactly a Hughes–Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^{n}}$ . We also prove that the MRD codes of minimum distance $2 in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

If G is a group with subgroup H and n, m are two fixed non-negative integers, we say that H is (n, m)-subnormal in G if there is a subgroup H0 containing H such that |H0:H|≤n and H0 is subnormal in G with subnormal defect at most m. We define S(n, m) to be the class of all groups of infinite special rank (and the trivial groups) whose subgroups of infinite special rank are (n, m)-subnormal in G. In this paper, we show that S(n, m)-groups are finite-by-nilpotent, in a large class of locally graded groups.

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X . It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite.

In each of the three projective planes coordinatized by the Knuth’s binary semifield K n of order 2 n and two of its Knuth derivatives, we exhibit a new infinite family of translation hyperovals. In particular, when n = 5 , we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied.

Let $\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\Omega|$. We obtain some properties of the set function $F:\Omega\mapsto \R^+$ defined by $$ F(\Omega)=\frac{T(\Omega)\lambda_1(\Omega)}{|\Omega|} ,$$ where $T(\Omega)$ and $\lambda_1(\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\'olya bound $F(\Omega)\le 1,$ and show that $$F(\Omega)\le 1- \nu_m T(\Omega)|\Omega|^{-1-\frac2m},$$ where $\nu_m$ depends only on $m$. For any $m=2,3,\dots$ and $\epsilon\in (0,1)$ we construct an open set $\Omega_{\epsilon}\subset \R^m$ such that $F(\Omega_{\epsilon})\ge 1-\epsilon$., Comment: 18 pages, 1 figure

The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

Let $k,n,m \in {\mathbb Z} ^{+}$ be integers such that $k\leq n \leq m$ , let $\mathrm {G}_{n,k}\in {\mathbb F} _{q^{m}}^{n}$ be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius $\tau $ , providing $\tau $ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in ${\mathbb F}_{q^{m}}^{n}$ . Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$ . As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when $n\mid m$ , we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +1$ , where $d$ is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +2$ , where $d$ is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of ${\mathbb F}_{q^{m}}^{n}$ of order $q^{kn}$ with $n$ dividing $m$ , such that $4n-3$ is a square in $\mathbb {Z}$ and containing $\mathrm {G}_{n,2}$ , is not efficiently list decodable at some values of the radius $\tau $ .

A groupGis called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to investigate the structure of uncountable groups of cardinality ℵ in which all proper subgroups of cardinality ℵ are metahamiltonian. It is proved that such a group is metahamiltonian, provided that it has no simple homomorphic images of cardinality ℵ. Furthermore, the behaviour of elements of finite order in uncountable groups is studied in the second part of the paper.

Let d , n ∈ Z + such that 1 ≤ d ≤ n . A d -code C ⊂ F q n × n is a subset of order n square matrices with the property that for all pairs of distinct elements in C , the rank of their difference is greater than or equal to d . A d -code with as many as possible elements is called a maximum d -code. The integer d is also called the minimum distance of the code. When d n , a classical example of such an object is the so-called generalized Gabidulin code (Gabidulin and Kshevetskiy (2005)). In Delsarte and Goethals (1975), Schmidt (2015) and K.U. Schmidt (2018), several classes of maximum d -codes made up respectively of symmetric, alternating and Hermitian matrices were exhibited. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric 2-code which is not equivalent to the one with same parameters constructed in Schmidt (2015).

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class $$\mathfrak {X}$$ is F-characteristic if any group containing an $$\mathfrak {X}$$-subgroup of finite index has also a characteristic subgroup of finite index that belongs to $$\mathfrak {X}$$. Thus the class $$\mathfrak {A}$$ of abelian groups is F-characteristic. The aim of this paper is to prove that many interesting classes of infinite groups are F-characteristic. Moreover, it is shown that the class of free groups and that of free abelian groups are not F-characteristic.

Extremal functions are exhibited in Poincare trace inequalities for functions of bounded variation in the unit ball $${\mathbb {B}}^n$$ of the n-dimensional Euclidean space $${{\mathbb {R}}}^n$$ . Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of $${\mathbb {B}}^n$$ , instead of just on $$\partial {\mathbb {B}}^n$$ , as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the median constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain $$\Omega \subset {{\mathbb {R}}}^n$$ , in terms of an isoperimetric inequality for subsets of $$\Omega $$ .