Anupam Singh
Let $G$ be an algebraic group of type $G_2$ over a field $k$ of characteristic $\neq 2,3$. In this paper we calculate centralizers of semisimple elements in anisotropic $G_2$. Using these, we show explicitly that there are six conjugacy classes of centralizers in the compact real form of $G_2$.
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Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.
The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.
Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralizers in G are conjugate within G. In this paper, we prove that the number of z-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least 6, and finite otherwise.
We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval.
Let $\mathscr{C}$ be a classical group defined over a finite field. We present comprehensive theoretical solutions to the following closely related problems: 1) List a representative for each conjugacy class of $\mathscr{C}$. 2) Given $x \in \mathscr{C}$, describe the centralizer $C_{\mathscr{C}}(x)$ of $x$ in $\mathscr{C}$, by giving its group structure and a generating set. 3) Given $x,y \in \mathscr{C}$, establish whether $x$ and $y$ are conjugate in $\mathscr{C}$ and, if they are, find explicit $z \in \mathscr{C}$ such that $z^{-1}xz = y$. We also formulate practical algorithms to solve th…
We explore when generator-conjugate homomorphisms are conjugate and when element-conjugate homomorphisms are conjugate from abelian or dihedral groups to the symmetric group. We completely determine when such homomorphisms are conjugate in the case where the source group has two generators by studying centralizer actions in the target group.
Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of $z$-classes in the unitary group over such fields is finite. Further, we count the number of $z$-classes in the finite unitary group $U_n(q)$, and prove that …
In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PSL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help us describe the stratification of the singular (orbifold) locus in some character varieties.
We describe the conjugacy classes of the elements of the free product of two groups and their centralizers and, as a consequence, we correct the calculation of the cyclic and periodic cyclic homology of the group ring of the free product of two groups given in a previous paper.
Verification of the conjugacy classes and ordinary character table of the Monster
本文在Monster群唯一性假设下，利用196883维忠实表示重新计算其共轭类与特征表，并验证了p<11的中心化子特征表及类融合。
A criterion for Tits alternative on the centralizer of a matrix
本文提出一个矩阵中心化子在GL(n,Z)中为多项式循环群的充要条件，即不包含非阿贝尔自由子群，并给出了使矩阵可交换的简单条件，从而在GL(n,Z)中建立有效的蒂茨替代准则，并应用于保持非退化有理双线性形式的算术群的共轭问题。
To classify the finite dimensional pointed Hopf algebras with $G= {\rm HS}$ or ${\rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}.
Computing centralizers in [f.g. free]-by-cyclic groups
本文证明有限生成自由-循环群中元素的中心化子可计算，并由此得出共轭元集合可计算及带上下文无关约束的共轭问题可判定，最后提出若干开放问题。
It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.
Let $G$ be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in $G$ of any element of $G$. Let $H$ be a subgroup of $G$. We give some conditions on $H$, under which we provide a solution to the conjugacy problem in $H$. We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$ and $\tilde{C}_n$.
We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.
A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result).…
The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elemen…
We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a conseque…
Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.
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.
Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checki…
Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
Counting conjugacy classes and automorphism-conjugacy classes of ZM-groups
本文研究ZM-群中元素的共轭类与自同构共轭类，给出了两类等价关系下的类数及各类大小的计数结果。
We characterize the group property of being with infinite conjugacy classes (or icc, i.e. in which all conjugacy classes beside 1 are infinite) for split extensions of groups.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
We characterise the group property of being with infinite conjugacy classes for wreath products of groups
We study quasi-semisimple elements of disconnected reductive algebraic groups over an algebraically closed field. We describe their centralizers, define isolated and quasi-isolated quasi-semisimple elements and classify their conjugacy classes.
We characterize the group property of being with infinite conjugacy classes (or icc, in which all conjugacy classes beside 1 are infinite) for extensions of some specific groups ; namely extensions of abelian, centerless, icc, or word hyperbolic groups.
Double coset problem for parabolic subgroups of braid groups
针对辫群中所有具有连通Coxeter图的抛物子群，首次给出双陪集问题的解，并推广至所有抛物子群。
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
We prove that any permutation group of degree $n \geq 4$ has at most $5^{(n-1)/3}$ conjugacy classes.
It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite
Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.
Let GL(n,q) be the group of nxn invertible matrices over a field with q elements, and SL(n,q) be the group of nxn matrices with determinant 1 over a field with q elements. We prove that the product of any two non-central conjugacy classes in GL(n,q) is the union of at least q-1 distinct conjugacy classes, and that the product of any two non-central conjugacy classes in SL(n,q) is the union of at least $\lceil\frac{q}{2} \rceil$ distinct conjugacy classes.
Products of conjugacy classes in $SL_2(R)$
本文针对SL₂(R)中的共轭类乘积问题，提出了计算任意n元组共轭类乘积的方法，解决了该代数结构下共轭类组合运算的计算问题
In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper.
We classify all finite groups G such that the product of any two non-inverse conjugacy classes of G is always a conjugacy class of G. We also classify all finite groups G for which the product of any two G-conjugacy classes which are not inverse modulo the center of G is again a conjugacy class of G.
For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
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。
We construct a hyperbolic group with a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.
Powers of conjugacy classes in a finite group
本文研究有限群中共轭类的幂为单个或两个共轭类之并对群可解性的影响，并证明这些条件可由字符表判定。
In this note, we determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also studied.
Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $α,β\in S_n$, we prove that the product $α^{S_n}β^{S_n}$ of the conjugacy classes $α^{S_n}$ and $β^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $α^{S_n}β^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $α,β\in S_n$ in the case when $α^{S_n}β^{S_n}$ is the union of exactly two distinct conjugacy classes.
Let A_n be the alternating group on n letters. For n>5, we describe the elements alpha, beta in A_n when alpha^{A_n} beta^{A_n} is the union of at most four distinct conjugacy classes.
We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).
Using the concept of algebraically closed groups, we prove that there is a countable torsion free group with exactly two conjugacy classes.
This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in particular, they are nilpotent of class at most $3$ and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.
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 $G$.
In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.
In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n is greater or equal to 3 and alternating group A_n, when n is greater or equal to 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S_n is determined by those restricted partitions of n-2 in which 1 and 2 do not appear as its part. And, in the case of alternating groups, it is determined by those restricted partitions of n-3 which has all its parts distinct, odd and in which 1 (and 2) does not appear as it…
In this paper, we study the structure of finite groups with a large number of conjugacy classes of $p$-elements for some prime $p$. As consequences, we obtain some new criteria for the existence of normal $p$-complements in finite groups.
Let $G$ be a nilpotent 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 $|a^G|=|b^G|=p$, where $p$ is an odd prime number. Set $a^G b^G=\{xy\mid x\in a^G, y\in b^G\}$. Then either $a^G b^G=(ab)^G$ or $a^G b^G$ is the union of at least $\frac{p+1}{2}$ distinct conjugacy classes. As an application of the previous result, given any nilpotent group $G$ and any conjugacy class $a^G$ of size $p$, we describe the square $a^G a^G$ of $a^G$ in terms of conjugacy classes of $G$.
We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$ are closed embedded submanifolds of $G$. We also prove that if $α:H\to G$ is a homomorphism of connected Lie groups such that the kernel of $α$ is discrete in $H$, then for an Ad-semisimple conjugacy class $C$ in $G$, every connected component of $α^{-1}(C)$ is a conjugacy class in $H$. Corresponding results for adjoint orbits in real Lie algebras are also proved.
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…
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。
Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by $\Cl(a)=\{gag^{-1}\mid g\in G\}$ the conjugacy class of $a$ in $G$. Assume that $|\Cl(a)|=p^n$. Then $\Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in \Cl(a^{-1})\}$ is the union of at least $n(p-1)+1$ distinct conjugacy classes of $G$.
We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.
A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these subgroups form a chain.
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