Profinite groups with pronilpotent centralizers

Profinite groups in which centralizers are abelian

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.

Profinite groups with restricted centralizers of $π$-elements

A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $π$, we take interest in profinite groups with restricted centralizers of $π$-elements. It is shown that such a profinite group has an open subgroup of the form $P\times Q$, where $P$ is an abelian pro-$π$ subgroup and $Q$ is a pro-$π'$ subgroup. This significantly strengthens a result from our earlier paper.

Commutators, centralizers, and strong conciseness in profinite groups

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form $[x_1,\dots,x_k]$, where $π(x_1)=π(x_2)=\dots=π(x_k)$. Here $π(x)$ denotes the set of prime divisors of the order of $x\in G$. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that $γ_k(G)$ is f…

On profinite groups in which centralizers have bounded rank

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p group, then G is virtually of rank at most r+1.

具有限制幂中心化的 profinite 群

Profinite groups with restricted centralizers of powers

研究具有有限指数或有限中心化的 profinite 群，证明其存在开正规子群使得商群具有有限指数。

有限群中的一致交换子群

Coprime commutators in profinite groups

本文研究了有限群中一致交换子群的性质，证明了若有限群的交换子群由可数多个procyclic子群覆盖，则其pronilpotent残余为有限-by-procyclic。

关于具有Magnus性质的投射有限群

On profinite groups with the Magnus Property

该研究探讨具有Magnus性质（MP）的投射有限群，证明其为投射可解群且商群、逆极限仍保持MP性质，推导其素因子范围与第二导子群性质，并证明有限生成的此类群必为有限群。

Strong conciseness of coprime and anti-coprime commutators

A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same primes. In the present paper we establish that a profinite group $G$ is finite-by-pronilpotent if the cardinality of the set of coprime commutators in $G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite commutator subgroup $G'$ if the cardinality of the set of anti-coprime commutators in $G$ is less than $2^{\aleph_0}$.

群的真可解完备化：实例与开放问题

The true prosoluble completion of a group: examples and open problems

群$Γ$的真预可解完备化$P\Cal S (Γ)$是$Γ$的可解子的投射系的逆极限。我们的目的是描述例子，并指出一些自然的开放问题。本文讨论了一个关于Profinite完备化的Grothendieck问题及其在真Prosoluble和真Pronilpotent完备化中的类似问题。1.导论. 2.关于有向正规子群集的完备化。3.泛性质。4.正规子群的有向集的例子。5.真正的prosoluble完成。6.例子. 7.关于Grothendieck问题的真prosoluble和真pronilpotent类似物。

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $γ_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $γ_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$…

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $φ$ of prime order such that every fixed point of $φ$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup. A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ a…

Profinite群的概率性质

Probabilistic properties of profinite groups

本文研究了profinite群在有限可解群或幂零群类下的$\mathfrak C$-positive性质，建立了等价条件并证明此类群为virtually prosoluble/nilpotent，同时给出了相关图的推论。

中心化子为阿贝尔群的紧致 $p$-adic 解析群

Compact $p$-adic analytic groups in which centralizers are abelian

本文利用多种代数与表示论方法构造了每个非平凡元素中心化子均为阿贝尔群的 pro-$p$ 群，回答了相关公开问题。

投射有限群Sylow子群的交换概率

Commuting probability for the Sylow subgroups of a profinite group

该研究探讨投射有限群的子群交换概率，证明若投射有限群的Sylow 2-子群与Sylow 3、5-子群的交换概率为正则群是虚投射可解的，还给出投射可解群为虚投射幂零的概率判据。

Profinite groups in which many elements have prime power order

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-groups yields a detailed description of profinite groups with that property. Then we deal with two generalizations of profinite CP-groups. Theorem 1.2. A profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. L…

On profinite groups with automorphisms whose fixed points have countable Engel sinks

An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if a profinite group $G$ admits an elementary abelian group of automorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each $a\in A\setminus\{1\}$ every element of the centralizer $C_G(a)$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ i…

A restricted Magnus property for profinite surface groups

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are …

好群的有限完备化中的有限子群

Finite subgroups of the profinite completion of good groups

本文证明剩余有限好群的有限 $p$-子群共轭类可自然双射对应到其有限完备化，并刻画中心化子与正规化子的闭包性质，且推广至有限可解子群并给出几何群论应用。

profinite群中互素换位子的强简明性

Strong conciseness of coprime commutators in profinite groups

本文定义了profinite群中的互素换位子$γ_j^*$和$δ_j^*$，并证明其值的集合基数小于$2^{\aleph_0}$当且仅当群具有特定的有限扩张结构。

Lannes的T-函子与 profinite 群的 mod-p 上同调

Lannes' $T$-functor and mod-$p$ cohomology of profinite groups

本文通过发展profinite空间参数化的离散扭模族乘积理论，证明了Lannes-Quillen定理对所有profinite群的完整版本，并应用于p- torsion元素和有限p-子群的共轭分离问题。

On groups with automorphisms whose fixed points are Engel

We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite gr…

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function $ζ_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value $ζ_G(k)^{-1}$ at a positive integer $k$ coincides with the probability that $k$ random elements generate the completed group ring of $G$. The explicit formulas obtained so far suggest that $ζ_G$ i…

On the procongruence completion of the Teichmüller modular group

For $2g-2+n>0$, the Teichmüller modular group $Γ_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $Π_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hatΠ_{g,n}$ its profinite completion. There is then a natural faithful representation $Γ_{g,n}\hookrightarrow Out(\hatΠ_{g,n})$. The procongruence completion $\checkΓ_{g,n}$ of the Teichmüller group is defined to be the closure of the Teic…