The groups and nilpotent Lie rings of order $p^8$ with maximal class

The Spherical Maximal Function on the Free Two-step Nilpotent Lie Group

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 $p$.

Maximal abelian normal subgroups in Nilpotent groups of class 2

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.

Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^m$, $q^{-s}$, and $q^{-ms}$ giving the ideal zeta function of the amalgamation of $m$ copies of $L$ over the derived subring, for every $m \geq 1$, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver rep…

Finite groups in which every maximal subgroup is nilpotent or normal or has $p'$-order

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 $q\neq p$.

Random Nilpotent Groups of Maximal Step

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 $\ell = n^2$.

Computing Nilpotent Quotients in Finitely Presented Lie Rings

A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor…

$ω$-stability and Morley rank of bilinear maps, rings and nilpotent groups

In this paper we study the algebraic structure of $ω$-stable bilinear maps, arbitrary rings and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

每个阶可被$p$整除的极大不变子群均为幂零群的有限群

Finite groups in which every maximal invariant subgroup of order divisible by $p$ is nilpotent

针对有限群$A$互素作用于有限群$G$的情形，对$|G|$的固定素因子$p$，完整刻画了所有阶可被$p$整除的极大$A$-不变子群均为幂零群的$G$的结构。

Nilpotent covers of symmetric and alternating groups

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…

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

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.

有限群中的p-幂零极大子群

p-Nilpotent maximal subgroups in finite groups

该研究针对每个极大子群要么p-幂零要么有素指数的有限群，证明其仅可涉入PSL_n(q)族中一种阶被p整除的非交换单群，且p=2时该单群必为特定参数的PSL_2(r^a)。

关于《库罗夫卡笔记本》中关于极大子群的两个问题

Finite groups with simple or $p$-nilpotent maximal subgroups

研究《库罗夫卡笔记本》中关于极大子群的问题，对于奇数\(p\)，证明满足特定条件的非单非可解群是\(p\)-幂零群和\(p\)-可分解群，\(p = 2\)时改进相关结果，给出特定非可解群的完整分类。

Classification of braces of cardinality $p^4$

We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent braces of the same cardinality, for sufficiently large $p$. These braces are explicitly obtained from the corresponding pre-Lie rings by the construction of the group of flows. Not right nilpotent braces of cardinality $p^4$ have been classified, hence our results finish the classification of braces of cardinality $p^4$.

后李环与斜括号的Lazard对应

A Lazard correspondence for post-Lie rings and skew braces

本文通过研究Lazard对应在李环半直和上的行为，建立了后李环与满足自然完备性条件的斜括号之间的对应关系，并推广了Smoktunowicz关于阶为$p^k$的斜括号与左幂零后李环对应的结果。

Skolem-Noether for nilpotent products

We consider the structure of groups and algebras that can be represented as automorphisms or derivations of distributive products -- which includes nonassociative rings, modules, forms, and commutation of groups and nonassociative loops. In many ways this is too broad to be possible, for instance even naive structure, such as inner automorphisms and derivations, are undefined for general bilinear maps. The appeal for working in this generality is broad applicability. We introduce fundamental structures and prove several results akin to ones in ring theory, including theorems of Morita and of S…

Compatible filters with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

某些特殊的非幂零极大不变子群具有素数幂指数的有限群

Finite groups in which some particular non-nilpotent maximal invariant subgroups have indices a prime-power

本文研究互素自同构群作用下有限群的结构，证明在特定极大不变子群为同构单群直积、且不含某正规子群的非幂零极大不变子群均具素数幂指数且排除PSL_2(7)作为合成因子的条件下，该群必可解。

Lie algebras with nilpotent length greater than that of each of their subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.

Completed representation ring spectra of nilpotent groups

In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[-] from groups to ring spectra, and show that the map R[G_p^] --> R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an a…

On the structure of cohomology rings of p-nilpotent Lie algebras

In this paper the authors investigate the structure the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose $E_{2}$-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen-Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifyi…

幂零群的环诱导局部化

Ring-induced localizations of nilpotent groups

研究交换环R诱导的群局部化函子，证明若R具有λ-环结构（如二项式环），则幂零群的R-局部化仍为幂零群，并给出新例子。

Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

We consider pro-isomorphic zeta functions of the groups $Γ(\mathcal{O}_K)$, where $Γ$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes $\mathfrak{p}$ of $K$ and depending only on the structure of $Γ$, the degree $[K : \mathbb{Q}]$, and the cardinality of the residue field $\mathcal{O}_K / \mathfrak{p}$. We show that the factors satisfy a certain uniform rationality and study their dependence on $[K : \mathbb{Q}]$. Explicit computation…

On central automorphisms of groups and nilpotent rings

Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ σ\mapsto h_σ$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_σ(x)=x^{-1} σ(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the…

类 $2$ 幂零群上的乘法 Lie 代数结构

Multiplicative Lie algebra structure on nilpotent group of class $2$

本文探讨了类 $2$ 幂零群上乘法 Lie 代数结构的性质，并提出了一种确定作为 Lie 环扩张的群（如亚循环群和类 $2$ 幂零群）上乘法 Lie 代数结构的方法。

Homotopy type of the nilpotent orbits in classical Lie algebras

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. In this paper we extend the above description of homotopy type of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.

Locally nilpotent linear groups

We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.

The topology of nilpotent representations in reductive groups and their maximal compact subgroups

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.

Gradings for nilpotent Lie 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.

Symbolic computation of Schur multipliers with an application to the groups of order dividing $p^6$

We describe an algorithm to compute the Schur multipliers of all nilpotent Lie $p$-rings in the family defined by a symbolic nilpotent Lie $p$-ring. Symbolic nilpotent Lie $p$-rings can be used to describe the isomorphism types of $p$-groups of order $p^n$ for $n \leq 7$ and all primes $p \geq n$. We apply our algorithm to compute the Schur multipliers of all $p$-groups of order dividing $p^6$.

On nilpotent subsemigroups in some matrix 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.

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that…

作为方程定义域的新群类

New classes of groups which are equational domains

该研究推广了非阿贝尔CSA群为方程定义域的结论，从k类幂零子群极大元非正规、极大局部幂零子群非正规两个方向，给出了非幂零/非局部幂零群成为方程定义域的新条件。

关于有限维 nilpotent 群和 Lie 环的上同调

On the cohomology of finite-dimensional nilpotent groups and Lie rings

研究有限维 nilpotent 群和 Lie 环的第一上同调群在不变子模为零时的消失性质，采用模型论方法，发展了基本代数方法，并应用于 Cartan 子环的 Frattini 论证和 definable 连接 p-可除交换群的 Maschke 定理变体。

Longer nilpotent series for classical unipotent subgroups

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $Θ(d^2)$ with nearly all factors ha…

Regular dessins uniquely determined by a nilpotent automorphism group

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…

On nilpotent Schur groups

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $\sym(G)$ that contains all right translations. We prove that every nonabelian nilpotent Schur group belongs to one of the explicitly given families of groups.

On cohomology rings of infinite groups

Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.