Manoj K. Yadav
Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of $M(G)$ and $F(M(G))$, where $F(M(G))$ denotes the Fitting subgroup of $M(G)$. We prove some conditional results regarding the nilpotency class of $M(G)$.
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This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
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 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.
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.
In this paper we introduce and study the poset of equivalence classes of subgroups of a finite group $G$, induced by the isomorphism relation. This contains the well-known lattice of solitary subgroups of $G$. We prove that in several particular cases it determines the structure of $G$.
In this paper we consider the {\em conjugacy stability} property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.
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.
In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |π(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and π(G) is the set of prime divisors of the group G.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.
In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.
We extend the notion of induced conjugacy classes in reductive groups, introduced by Lusztig and Spaltenstein for unipotent classes, to arbitrary classes. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of canonical parabolic subgroups attached to elements of reductive groups in the sense of Jacobson-Morozov. These results are prerequisites for making the geometric expansion of the Arthur-Selberg trace formula more explicit.
Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $Σ$ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $Σ\subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $Σ$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Σ$ is a $G$-covering subgroup system for the classes of all $σ$-soluble and all $σ$-nilpotent groups, and for the class of all $σ$-soluble $PσT$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Ko…
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $η(G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $η(G) \ge n-2$, and we determine when equality holds.
Triangles in the graph of conjugacy classes of normal subgroups
研究有限群正规子群共轭类图无三角形或恰有一个三角形时的结构，给出了相应的刻画。
Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup $G'$ is finite. We establish the following results. Let $K,n$ be positive integers and $G$ a group having a $K$-approximate subgroup $A$. If $|a^G|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle A^G\rangle$ has finite $(K,n)$-bounded order. If $|[g,a]^G|\leq n$ for all $g\in G$ and $a\in A$, then the commutator subgroup of $[G,A]$ has fin…
We give another proof of an observation of Thévenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.
On Posets of Classes of Automorphic Subgroups of Finite Groups
本文研究了有限群的自同构子群类偏序集AutCl(G)，通过引入偏序解决了Tarnauceanu提出的问题5，并证明了AutCl(Dn)和AutCl(Q4m)是分配格，同时刻画了使AutCl(G)构成链的所有有限群类。
Landau's Theorem on conjugacy classes for normal subgroups
本文推广Landau定理，证明对任意正整数n和s，具有指数为n且含s个非中心G-共轭类的正规子群的有限群在同构意义下仅有有限多个，并给出阶的上界及GAP分类结果。
Conjugacy classes of completely reducible cube-free solvable p'-subgroups of GL(2, q)
本文针对无立方因子正整数m与不整除m的素数p，确定了一般线性群GL(2, q)（q为p的幂）中阶为m的完全可约可解无立方因子子群的共轭类数量。
A note on finite groups with few conjugacy classes of subgroups
该研究基于有限群\(G\)的子群共轭类数量\(Con(G)\)给出其可解性条件，对\(Con(G)\leq5\)的有限群分类，并给出\(A_5\)新特征。
Groups with 5 nontrivial conjugacy classes of non self-normalizing subgroups are solvable
本研究针对Bianchi等人提出的关于$\mathcal{D}_5$（恰有5个非平凡非自正规子群共轭类的有限群族）由导出长度不超过3的可解群组成的猜想，完成验证并证实该猜想成立。
Conjugacy classes of completely reducible cyclic subgroups of GL$(2, q)$
本文针对素数幂$q$及不被素数$p$整除的正整数$m$，确定了一般线性群GL$(2,q)$中阶为$m$的完全可约循环子群的共轭类数目。
Finite groups with few conjugate classes of minimal non-abelian subgroups
本文研究了有限非交换群中极小非交换子群的共轭类数量，确定了当该类数为1时群的结构，并在p-群情形下给出了类数不超过p时的结构。
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…
It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
On Posets of Classes of Subgroups with Same Set of Orders of Elements
本文研究了有限群中具有相同元素阶集的子群类的偏序集，证明其在p群中为链，并探讨了该偏序集在循环群和二面体群中的格结构及其分配和模特性。
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。
Conjugacy classes of $π$-elements and nilpotent/abelian Hall $π$-subgroups
本文研究有限群中 π-元素共轭类的数量下界，并证明当该数量足够大时，群必含有幂零或阿贝尔的 Hall π-子群。
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup.
We explore some of the special features with respect to Bredon cohomology of groups having all its finite subgroups either nilpotent or p-groups or cyclic p-groups. We get some results on dimensions and also a formula for the equivariant Euler class for certain groups. We consider the generalization for Bredon cohomology of the properties of being duality or Poincaré duality and study their behavior under p-power index extensions with coefficients in a field of characteristic p.
We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup.
We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.
Classes of Subgroups of a finite group with isomorphic quotient groups
本文研究有限群中具有同构商群的子群类构成的偏序集QIso(G)，针对阿贝尔群及几类非阿贝尔亚循环群展开分析，得出QIso(G)的同构性质及相关群类的刻画结论。
In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $Γ(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $ΔP_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.
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.
On finite groups in which the twisted conjugacy classes of the unit element are subgroups
该研究探讨扭共轭相关子群性质的群结构，构造出任意幂零类的有限满足群及无限非幂零满足群，证否了既有文献中的相关猜想。
Supersolubility of a finite group $G=\langle A,B\rangle$ with the nilpotent derived subgroup $G^\prime$ is established under the condition that the subgroups $A$ and $B$ are both subnormal and supersoluble.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.
We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary of this theorem says that a noncyclic free group of finite rank admits a faithful highly transitive action on an infinite set, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.
Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less tha…
We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.
Detecting solvability, supersolvability and CLT properties via an invariant based on conjugacy classes of subgroups
该研究通过定义基于子群共轭类的不变量d*(G)，推导得出d*(G)与有限群G的可解性、超可解性及CLT性质相关的判定准则。
We prove that every verbally closed two-generated subgroup of a free solvable group G of a finite rank is a retract of G.
For any finite group Q not of prime power order, we construct a group G that is virtually of type F, contains infinitely many conjugacy classes of subgroups isomorphic to Q, and contains only finitely many conjugacy classes of other finite subgroups.
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。
Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine completely when this subgroup is finitely generated, by using preceding results on centralizers of reflections. This result provides a new and naturally constructed example of a family of non-finitely generated subgroups of finitely generated groups.
We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal subgroup isomorphic to the multiplicative subgroup of Q.
We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.
Let $G$ be a finite group and $H$ be a subgroup of $G$. In this paper, we prove that if $G$ is a finite nilpotent group and $H$ a subgroup of $G$, then $H$ is normal in $G$ if and only if all normalized right transversals of $H$ in $G$ are isotopic, where the isotopism classes are formed with respect to induced right loop structures. We have also determined the number isotopy classes of transversals of a subgroup of order 2 in $D_{2p}$, the dihedral group of order $2p$, where $p$ is an odd prime.
Finite groups can be generated by a pi-subgroup and a pi'-subgroup
该研究回答了Dan Haran的问题，推广了前人相关结果，证明任意有限群可由一个π-子群和一个π'-子群生成，并给出可数生成自由投射有限群的自由积描述。
We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple $\QQ $-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.
In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong C_3*C_2$ are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers $p\ge 3$, $q\ge 2$ the triangle group $Γ=Δ(p,q,\infty)\cong C_p*C_q$ has uncountably many conjugacy classes of nonparabolic maxima…
We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for …
Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.
The spectral space of conjugacy classes of subgroups of a compact Lie group
该研究指出紧李群子群共轭类的Zariski拓扑空间可分解为闭开块，各块由有限Weyl群子群主导且结构由分支群整表示控制，揭示了其拓扑结构特性。
Let $V$ be a vector space over the field of order $2$. We investigate subgroups of the linear group $GL(V)$ which are generated by a conjugacy class $D$ of elements of order $3$ such that all $d$ in $D$ have $2$-dimensional commutator space $[V,d]$.
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results e…
We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.
On the structure of finitely generated subgroups of branch groups
本文描述了具有子群归纳性质的分支群（如第一Grigorchuk群和挠GGS群）有限生成子群的块结构。
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