Conjugacy classes of $p$-elements and normal $p$-complements

连接共轭类与正规子群的最小不变特征标

Linking conjugacy classes and minimal invariant characters of normal subgroups

研究有限群G的正规子群N的最小G-不变特征标（即G在Irr(N)上共轭作用轨道上的特征标和）对N结构的影响及其与N的G-共轭类的关系。

On the regularity of a graph related to conjugacy class sizes of a normal subgroup

Given a finite group $G$ with a normal subgroup $N$, the simple graph $Γ_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$ containing the element $x$. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not co-prime. In this article we prove that, if $Γ_G(N)$ is a connected incomplete regular graph, then $N= P \times{A}$ where $P$ is a $p$-group, for some prime $p$ and $A\leq{Z(G)}$, and ${\bf Z}(N)\not = N\cap {\bf Z}(G)$.

有限群共轭类大小限制下正规p-补的存在性

On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes

本文证明了在共轭类大小满足特定限制条件下，有限群存在正规p-补。

与有限群中正规子群的共轭类相关的图

Graphs associated to conjugacy classes of normal subgroups in finite groups

本文研究有限群正规子群对应的两类共轭类图，证明其连通分量数至多2，给出直径上界并刻画不连通时的群结构。

超可解补群关于阿贝尔基底的共轭条件及非互素作用的不动点结果

Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions

本文研究了有限分裂扩张中阿贝尔基底的超可解补群的共轭条件，证明了基于 Sylow $p$-子群共轭性的充要条件，并放宽了 Higman 的限制，最后得到了一个类似 Glauberman 引理的非互素作用不动点结果。

On the conjugacy problem for cyclic extensions of free groups

We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given.

Products of conjugacy classes in simple algebraic groups in terms of diagrams

For a simple algebraic group $G$ over an algebraically closed field, we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of $G$. We use two types of labels, a binary marking and a labeling with non-negative integers. The first is used to recognize large conjugacy classes which appear in a product of two conjugacy classes while the second is used to keep track of multiplicities of regular diagrams. In particular, we formulate sufficient conditions in terms of marked diagrams, for a product of normal subsets in $G$ to contain regular semisimple elements.

正规子群的陪集与两个共轭类的并集

Cosets of normal subgroups and union of two conjugacy classes

研究有限群正规子群陪集包含于两个共轭类并集的条件，证明此时 $N$ 不必可解但结构受限，其非交换主因子同构于奇特征 Lie 型单群的直积。

On the conjugacy growth functions of groups

To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in "normal" cases, groups with exponential growth functions also have exponential conjugacy growth functions.

对称群与交错群的共轭类平铺

Tilings of symmetric and alternating groups by conjugacy classes

本文研究对称群与交错群被共轭类平铺的问题，证明除对换类外所有非单位共轭类均不能平铺对称群，且交错群中避开单位元的正规子集不能平铺。

有限群中共轭类乘积的若干问题

Some problems about products of conjugacy classes in finite groups

本文总结了与共轭类乘积或幂中类数量相关的有限群非单性、可解性及正规结构的结果，并收集了部分尚未完全解决的问题。

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let $G$ be a finite group and $A$ be a normal subgroup of $G$. We denote by $ncc(A)$ the number of $G$-conjugacy classes of $A$ and $A$ is called $n$-decomposable, if $ncc(A)=n$. Set ${\cal K}_G = \{ncc(A)| A \lhd G \}$. Let $X$ be a non-empty subset of positive integers. A group $G$ is called $X$-decomposable, if ${\cal K}_G = X$. Ashrafi and his co-authors \cite{ash1,ash2,ash3,ash4,ash5} have characterized the $X$-decomposable non-perfect finite groups for $X = \{1, n \}$ and $n \leq 10$. In this paper, we continue this problem and investigate the structure of $X$-decomposable non-perfect fi…

Homogeneous products of conjugacy classes

Let $G$ be a finite group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that ${\bf C}_G(a)={\bf C}_G(b)$. Then $a^G b^G$ is a conjugacy class if and only if $[a,G]=[b,G]=[ab,G]$ and $[ab,G]$ is a normal subgroup of $G$.

On conjugacy separability of fibre products

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in…

交错群中共轭类的平方

Squares of Conjugacy Classes in Alternating Groups

本文将对称群中共轭类平方覆盖非平凡元素的结论推广至交错群，给出相应条件。

On subgroup conjugacy separability in the class of virtually free groups

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.

Alternating groups as products of four conjugacy classes

Let $G$ be the alternating group $\mbox{Alt}(n)$ on $n$ letters. We prove that for any $\varepsilon > 0$ there exists $N = N(\varepsilon) \in \mathbb{N}$ such that whenever $n \geq N$ and $A$, $B$, $C$, $D$ are normal subsets of $G$ each of size at least $|G|^{1/2+\varepsilon}$, then $ABCD = G$.

有限群中实共轭类的平方

Squares of real conjugacy classes in finite groups

本文研究有限群中实共轭类平方的结构，证明满足特定平方形式的共轭类生成可解正规子群，并完全刻画其结构及相应群类。

Fixed conjugacy classes of normal subgroups and the k(GV)-problem

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply these formulas effectively, the parameter c, which in general is hard to control, is studied in some important situations. These results are then used to provide a new, shorter proof of the most difficult case of the well-known k(GV)-problem, which occurs for p=5 and V induced from the natural module of a 5-complement of GL(2,5). We also show how, for large p, …

Conjugacy classes of maximal cyclic subgroups

In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $η(G/N) = η(G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomo…

Simultaneous Conjugacy Classes of Finite $p$-groups of rank $\leq 5$

For a finite group $G$, we consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. This paper concern studied of normalized functions $A_G(t/|G|)$ and $B_G(t/|G|)$ for finite $p$-groups of rank at most $5…

On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.

A conjugacy criterion for Hall subgroups in finite groups

A finite group $G$ is said to satisfy $C_π$ for a set of primes $π$, if $G$ possesses exactly one class of conjugate $π$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_π$ in terms of a normal series of the group.

On the conjugacy problem in group $\bf F/{N_1\cap N_2}$

Let $N_1$ (resp., $N_2$) be the normal closure of a finite symmetrized set $R_1$ (resp., $R_2$) of a finitely generated free group $F = F(A)$. It is well-known that if $R_i$ satisfies the condition C(6), then the conjugacy problem is solvable in $F/N_i$. In the present paper we prove that if $R_1\cup R_2$ satisfies the condition C(6) and the presentation $<A\, \mid R_1,R_2>$ is atorical, then the conjugacy problem is solvable in $F/{N_1\cap N_2}$. In particular, if $R_1\cup R_2$ satisfies the condition C(7) then the conjugacy problem is solvable in $F/{N_1\cap N_2}$.

可解群中的局部共轭性

Local conjugacy in prosolvable groups

本文研究可解群中局部共轭补子的共轭条件，通过第一上同调集的主型分解给出证明，并推广了有限群的结果，同时提供子群包含的局部条件和定点结果。

共轭类在陪集中的特征公式

Character formula for conjugacy classes in a coset

本文研究了有限群中共轭类与不可约特征标之间的关系，特别探讨了在循环群情况下陪集中的共轭类数量与特征标扩展的关系。

Conjugacy Distinguished Subgroups

Let ${\cal C}$ be a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal C}$-distinguished if whenever $y\in R$, then $y$ has a conjugate in $H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R/N\in {\cal C}$ of $R$. We prove that in a group having a normal free subgroup $Φ$ such that $R/Φ$ is in ${\cal C}$, every finitely generated subgroup is conjugacy ${\cal C}$-distinguished. We also prove that finitely generated sub…

On groups where the twisted conjugacy class of the unit element is a subgroup

We study groups $G$ where the $φ$-conjugacy class $[e]_φ=\{g^{-1}φ(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $φ$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.

Conjugacy classes of p-torsion in symplectic groups over S-integers

For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes of subgroups $P$ of order $p$ in the symplectic group and the ideal class group in the ring $Z[1/n]$. This is used for the study of these classes. In particular we determine the centralizer $C(P)$ and $N(P)/C(P)$ where $N(P)$ denotes the normalizer.

具有循环极大子群的非阿贝尔有限p-群的共轭直径

The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups

本文确定了半二面体2-群、广义四元数群和模p-群的共轭直径，推广了二面体群的已有结果。

Conjugacy classes and straight elements in Coxeter groups

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called…

具有互质大小共轭类的群

Groups with conjugacy classes of coprime sizes

针对有限群中位于互质大小共轭类的元素，证明其共轭类生成子群的交为阿贝尔正规子群，并推广$π$-正则元共轭类乘积的性质，扩展了$p$-正则共轭类公因子图的相关结果。

Simultaneous Conjugacy Classes as Combinatorial Invariants of Finite Groups

Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. We show that $A_G(t)$ determines and is completely determined by the class equation of $G$. We show that $α_{G,n}$ gro…

关于 $\ell$-正则共轭类的数量

On the number of $\ell$-regular conjugacy classes

本文证明了当素数 $\ell>5$ 时，有限群的 $\ell$-正则共轭类数不少于其 Sylow $\ell$-子群正规化子的相应数量。

On finitely generated normal subgroups of right-angled Artin groups and graph products of groups

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …

Finite simple groups with short Galois orbits on conjugacy classes

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.

Normal form and parabolic dynamics for quadratically growing automorphisms of free groups

We present a normal form for outer automorphisms $ϕ$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $ϕ$). In analogy to the known normal form for linearly growing automorphisms as efficient Dehn twist, our normal form for $ϕ$ is given in terms of a 2-level Dehn twist on a graph-of-groups $\cal{G}$ with $π_1 {\cal{G}} \cong F_N$, where a conjugacy class of $F_N$ grows at most linearly if and only if it is contained in a vertex group of $\cal{G}$. Our proof is based on earlier work of the second aut…

Power-associative, conjugacy closed loops

We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If $Q$ is a PACC-loop with nucleus $N$, then $Q/N$ is an abelian group of exponent 12; if in addition $Q$ is finite, then $|Q|$ is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops of order 27, four of which have the automorphic inverse property. We also study some special elements in loops, such as Moufang elements, weak inverse property (WIP) elements, and extra elements. In a CC-loop, the set of W…

具有大量$p$-正则共轭类的有限群

Finite groups with many $p$-regular conjugacy classes

该文研究具有大量$p$-正则共轭类的有限群结构，给出保证群$p$-可解的精确下界、奇素数下紧的$p$-长度界及正规Sylow $p$-子群存在的最优判别准则。

A Polynomial Time Algorithm For The Conjugacy Decision and Search Problems in Free Abelian-by-Infinite Cyclic Groups

In this paper we introduce a polynomial time algorithm that solves both the conjugacy decision and search problems in free abelian-by-infinite cyclic groups where the input is elements in normal form. We do this by adapting the work of Bogopolski, Martino, Maslakova, and Ventura in \cite{bogopolski2006conjugacy} and Bogopolski, Martino, and Ventura in \cite{bogopolski2010orbit}, to free abelian-by-infinite cyclic groups, and in certain cases apply a polynomial time algorithm for the orbit problem over $\Z^n$ by Kannan and Lipton.

具有大小为$2p^n$的共轭类的有限群

Finite groups with conjugacy classes of size $2p^n$

该研究推广了Burnside $p^a$定理，证明非阿贝尔有限单群无大小为$2p^n$的共轭类，并推广Kazarin结果，明确有限群中特定大小共轭类生成子群的可解性及非可解时的结构。

Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear grou…

Conjugacy classes in maximal parabolic subgroups of general linear groups

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of …

共轭类的平方与 Baer-Suzuki 定理的一个变体

Squares of conjugacy classes and a variant on the Baer-Suzuki Theorem

本文针对有限群中阶为 $p$ 的元素构成的正规子集 $A$，证明了若 $A^2$ 均为 $p$-元素则生成子群可解，并在 $O_p(G)=1$ 条件下确定了相关群结构性质，同时给出了最优性反例。

Canonical Symplectic Representations for Prime Order Conjugacy Classes of the Mapping-class Group

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an {\sl adapted} homotopy basis and there is a corresponding {\sl adapted presentation}. We also give a necessary and sufficient condition for a prime order symplectic matrix to be the image of a prime order element in the mapping-class group.

On Normalized Table Algebras Generated by a Faithful Non-real Element of Degree 3-II

The concept of "table algebra" was introduced by Z Arad anf H. Blau in order to study in a uniform way properties of products of conjugacy classes and of irreducible characters of a finite group, Except for certain cases which remain open, Normalized Table Algebras Generated by a Faithful Non- real Element of degree 3. As application we classified finite groups with a faithful non-real irreducible character of dimension 3 without using character theory of finite groups.

由 $p$-正则共轭类大小的不连通公因子图得到的可解性

Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes

本文证明若有限群 $G$ 的 $p$-正则元素共轭类大小的公因子图不连通，则 $G$ 可解，通过构造稳定化子并利用 Dolfi-Lucido 定理与 Burnside 定理排除非阿贝尔合成因子。

交换子的阶与共轭类乘积在有限群中的研究

Orders of commutators and Products of conjugacy classes in finite groups

本文研究了有限群中交换子的阶与共轭类乘积的性质，证明了在特定条件下共轭类生成的子群是可解的。

Special involutions and bulky parabolic subgroups in finite Coxeter groups

The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

二面体Artin群中的扭曲共轭 I：环面纽结群

Twisted conjugacy in dihedral Artin groups I: Torus Knot groups

本文针对奇数二面体Artin群的扭曲共轭问题，提出一种基于环面纽结群替代表示和测地线正规形式的线性时间算法，并证明该结果可应用于扩张群中的共轭问题。

计算${\ m PSL}_2(p)$的子群共轭类

Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$

该研究推导了${\ m PSL}_2(p)$（p为素数）非单位真子群的同构类、共轭类及两类自正规相关共轭类的计数公式，证明了p>37时的四类下界，结合数值搜索与数论结果给出了下界可达性与上界结论。

Conjugacy in Garside groups I: Cyclings, powers, and rigidity

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||Δ||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This s…

On solvable groups with one vanishing class size

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylo…

On zeros of irreducible characters lying in a normal subgroup

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.