We develop a general approach to the study of maximal nilpotent subsemigroups of finite semigroups. This approach can be used to recover many known classifications of maximal nilpotent subsemigroups, in particular, for the symmetric inverse semigroup, the symmetric semigroup, and the factor power of the symmetric group. We also apply this approach to obtain a classification of maximal nilpotent subsemigroups in the semigroup of binary relations and in certain 0-simple finite semigroups.
We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup $Ω_n$ of all $n\times n$ real matrices with non-negative coefficients and the semigroup $\mathbf{D}_n$ of all doubly stochastic real matrices.
All isolated, completely isolated, and nilpotent subsemigroups in the semigroup $\IS$ of all injective partial transformations of an $n$-element set, considered as a semigroup with a sandwich multiplication are described.
We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids. Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining…
We study the structure of nilpotent subsemigroups in the semigroup $M(n,\mathbb{F})$ of all $n\times n$ matrices over a field, $\mathbb{F}$, with respect to the operation of the usual matrix multiplication. We describe the maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency degree and classify them up to isomorphism. We also describe isolated and completely isolated subsemigroups and conjugated elements in $M(n,\mathbb{F})$.
We give a description of finite semigroups $S$ that are minimal for not being Malcev nilpotent, i.e. every proper subsemigroup and every proper Rees factor semigroup is Malcev nilpotent but $S$ is not. For groups this question was considered by Schmidt.
In this paper we describe a portion of the subsemigroup lattice of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the countable infinite set $Ω$. Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing the symmetric group $\sym(Ω)$. The portion of the subsemigroup lattice of $Ω^Ω$ which we describe is that between the intersection of these five maximal subsemigroups and $Ω^Ω$. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$ subsemigroups between $\sym(Ω)$ and $Ω^Ω$.
We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X=N of natural numbers containing a given subsemigroup W of T(X) where each element of a given set $U$ is a generator of T(X) modulo W. This note continue the study of maximal subsemigroups on the monoid of all full transformations on a infinite set.
We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that $S_n$ consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, and compute nilpotency indexes. Furthermore, we give a necessary and sufficient condition for the $j$…
In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. …
We consider here the free two step nilpotent Lie group, provided with the homogeneous Korányi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.
A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free nilpotent group of class 2 such a group is always a cyclic extension of the center. However this need not be true in an arbitrary finitely generated class two nilpotent group. We show that three generator nilpotent groups violating this condition can be easily found. However in the case when there are four or more generators certain conditions must hold.
Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) There exists at most one prime divisor $q$ of $|G|$ such that $G$ is neither $q$-nilpotent nor $q$-closed, where $q\neq p$.
Let $G$ be a random torsion-free nilpotent group generated by two random words of length $\ell$ in $U_n(\mathbb{Z})$. Letting $\ell$ grow as a function of $n$, we analyze the step of $G$, which is bounded by the step of $U_n(\mathbb{Z})$. We prove a conjecture of Delp, Dymarz, and Schafer-Cohen, that the threshold function for full step is $\ell = n^2$.
We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of $\mathcal{M}$ is equal to the order of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the…
We classify the nilpotent Lie rings of order $p^8$ with maximal class for $p \ge 5$. This also provides a classification of the groups of order $p^8$ with maximal class for $p \ge 11$ via the Lazard correspondence.
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
Starting from the symmetric group $S_n$, we construct two fiat $2$-categories. One of them can be viewed as the fiat "extension" of the natural $2$-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other $2$-category can be viewed as the fiat "extension" of the $2$-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an o…
Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γgenerated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to prove that there is a strong deformation retraction of Hom(Γ,G) onto Hom(Γ,K). We also obtain a strong deformation retraction of the geometric invariant theory quotient Hom(Γ,G)//G onto Hom(Γ,K)/K.
Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity $ω_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset of $L$. In the case that $L$ is a finite group of Lie type, we identify covers of $L$ by $c$-nilpotent subgroups, and we use these covers to constru…
For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that MaxDim$(G)=m(G)$ if the derived subgroup of $G$ is nilpotent. However MaxDim$(G)-m(G)$ can be arbitrarily large: for any odd prime $p,$ we construct a finite soluble group with Fitting length 2 satisfying $m(G)=3$ and MaxDim$(G)=p.$
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.
We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing $\ell^{q,p}$ cohomology.
The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$…
