Daniele Garzoni, Nick Gill
Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)<n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$, or $G$ belongs to explicit families of examples.
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Some nonabelian subgroups of the Nottingham group over F_4
本文分类了 F_4 上诺丁汉群中阶为 8 的最小分歧非交换子群的共轭类，并利用有限自动机显式构造了各类代表元。
The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups $H$ of finite quasiprimitive permutation groups $G$. If the socle of $G$, denoted by $\mbox{soc}(G)$, is nonabelian, then $H$ lies in $\mbox{soc}(G)$. An explicit description is given for the possibilities of $H$ under the condition that $H$ does not contain a nontrivial normal subgroup of $\mbox{soc}(G)$.
Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.
Conjugacy classes with covering number two in hyperbolic orthogonal groups
本文针对双曲二次向量空间的正交群换位子群，证明其存在覆盖数相关的共轭类，为Thompson关于有限非阿贝尔单群共轭类的猜想提供了相关群论结果。
We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of $G$ by the solvable radical of $G$ is an almost simple group. The main goal of this paper is prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when $G$ is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime…
For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. Recently the following question has been asked: Is it true that for each nonabelian finite simple group $S$ and each $n\in\mathbb{N}$, if the set of class sizes of a finite group $G$ with trivial center is the same as the set of class sizes of the direct power $S^n$, then $G\simeq S^n$? In this paper we approach an answer to this question by proving that $A_6\times A_6$ is uniquely determined by $N(A_6\times A_6)$ among finite groups with trivial center.
Generation by conjugate elements of finite almost simple groups with a sporadic socle
本文研究散在基座有限几乎单群中自同构的共轭元生成问题，通过特征标表计算细化Guralnick–Saxl估计，证明除Fischer群3-对换情形外α_S(x)≤4，并给出例外群的精确值。
On generations by conjugate elements in almost simple groups with socle $\mbox{}^2F_4(q^2)'$
本文证明了基座为 $\mbox{}^2F_4(q^2)'$ 的几乎单群可由四个共轭元素生成，并利用此结果证实了关于有限群 $\pi$-根的一个猜想对特定非交换合成因子类型的群成立。
We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.
In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of the classical socle and $q$ is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on class numbers, minimal …
The Moebius function on the lattice of normal subgroups
本文通过研究正规子群格，特别是基座和根的格，获得了生成群所需共轭类最小数目的表达式，证明该数目可由特征标表刻画，并利用莫比乌斯函数提取了群忠实不可约表示的信息。
Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial c…
Spectrum of conjugacy and order super commuting graphs of some finite groups
该研究定义了等价关系下的超图结构，证明其为完全图的广义并，进而推导三类有限群的共轭与阶超交换图的邻接谱和拉普拉斯谱。
A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an …
Several key agreement protocols are based on the following "Generalized Conjugacy Search Problem": Find, given elements b_1,...,b_n and xb_1x^{-1},...,xb_nx^{-1} in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group B_N, Hughes and Tannenbaum suggested a length-based approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated. We introduce several effective realizations of this approach. In particular, a new length function is defined on B_N which possesses significantly better properties than the natural le…
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。
On fixed-point-free involutions in actions of finite exceptional groups of Lie type
本文研究了有限例外李型群作用中无固定点反置映射的分类问题，确定了具有2-elusive socle的几乎简单原始例外群的结构。
On the Borel complexity of the space of left-orderings of nilpotent groups
本文首次构造非交换左序群，其序空间共轭轨道关系有无穷轨道但Borel光滑，并给出幂零群光滑性的充分条件，同时证明幂零性非充分条件。
We analyze the structure of the \emph{frequency space} $Q(F)$ of a nonabelian free group $F=F(a_1,...,a_k)$ consisting of all shift-invariant Borel probability measures on $\partial F$ and construct a natural action of $Out(F)$ on $Q(F)$. In particular we prove that for any outer automorphism $ϕ$ of $F$ the \emph{conjugacy distortion spectrum} of $ϕ$, consisting of all numbers $||ϕ(w)||/||w||$, where $w$ is a nontrivial conjugacy class, is the intersection of $\mathbb Q$ and a closed subinterval of $\mathbb R$ with rational endpoints. We also provide an algorithm for detecting strict hyperboli…
Fixers and derangements of finite permutation groups
本文对以 PSL_2(q) 为基座的几乎单本原群的大固定元子群进行共轭分类，并验证了 Spiga 猜想的一个特例，同时给出交错群和散在单群情形的相关结果。
Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.
In noncommutative geometry a `Lie algebra' or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset C stable under inversion. We study the associated Killing form. For the universal calculus associated to C=G \ {e} we show that the magnitude of the Killing form μ=\sum_{a,b\in C}K^{-1}_{a,b} is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ is not equal to 1/(N-1), where N is the number of conjugacy classes…
Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.
Let $G$ be a nontrivial transitive permutation group on a finite set $Ω$. An element of $G$ is said to be a derangement if it has no fixed points on $Ω$. From the orbit counting lemma, it follows that $G$ contains a derangement, and in fact $G$ contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that $G$ is almost elusive if it contains a unique conjugacy class of derangements of…
Prime power coverings of groups
本文证明了在先天传递作用（含极大子群情形）下，有限群 $A$-素数幂覆盖子群猜想，即 $|G:U|$ 被 $|A:G|$ 的函数界定，并给出了非交换有限单群阶的新界。
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $π_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class…
Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the natural action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$. We show that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group, that is, for every $ϕ,ψ\in \mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$, if there exists $k \in \mathbb{N}^*$ such that $ϕ^k=ψ^k$, then $ϕ=ψ$. This answers a question of Handel and Mosher. We then use the fact that $\mathr…
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