Yu Zeng, Dongfang Yang, Silvio Dolfi
We prove that the function $\mathrm{P}_{\mathrm{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes very sparse values in a large segment of the $[0,1]$ interval.
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In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of vanishing elements attain this bound. For symmetric groups of degree at least $5$, we show that this bound is at least $2327/2520$ which is best possible.
Finite groups with a small proportion of vanishing elements
本文对消失元素比例函数取值 $\frac{m-1}{m}$（$m=1$ 至 $6$）的有限群进行了完全分类，完善了此前关于该函数在区间 $[0, \mathrm{P}_{\mathbf{v}}(A_7))$ 内取值受限的结果。
Many results have been established about determining whether or not an element evaluates to zero on an irreducible character of a group. In this note it is shown that if a group $G$ has a normal nilpotent subgroup $N$, and $P$ is a Sylow $p$-subgroup of $G$, then no irreducible character of $G$ vanishes on $N\cap Z(P)$.
Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.
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…
For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way, we also provide new upper bounds and answer some open questions of the first author, Pálfy and Saxl.
Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $χ$ of $G$ such that $χ(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group.
For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. …
Non-vanishing elements and complex group algebras
本文研究有限群的特征标理论，证明对每个有理非零化元素 $x$，其中心化子 $C_G(x)$ 的阶至少为群权重 $\omega(G)$。
Let $ G$ be a finite group and $p$ be a prime. Let $ \mathrm{Vo}(G) $ denote the set of the orders of vanishing elements, $\mathrm{Vo}_{p} (G)$ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements divisible by $p$ and $\mathrm{Vo}_{p'} (G) $ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements not divisible by $p$. Dolfi, Pacifi, Sanus and Spiga proved that if $ a $ is not a $ p $-power for all $ a\in \mathrm{Vo}(G)$, then $ G $ has a normal Sylow $ p $-subgroup. In another article, the same authors also show that if if $ \mathrm{…
Let $ x $ be an element of a finite group $ G $ and denote the order of $ x $ by $ \mathrm{ord}(x) $. We consider a finite group $ G $ such that $ \gcd(\mathrm{ord}(x),\mathrm{ord}(y))\leqslant 2 $ for any two vanishing elements $ x $ and $ y $ contained in distinct conjugacy classes. We show that such a group $ G $ is solvable. When $ G $ with the property above is supersolvable, we show that $ G $ has a normal metabelian $ 2 $-complement.
In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of |G| distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of (1 - 1/p)/e, where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain "large" subspaces of the natural G-module. We find explicit upper and lower bounds for the propor…
In a paper by the first author it was shown that for certain arithmetical results on conjugacy class sizes it is enough to only consider the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.
A new characterization of $E_8 (p)$ via its vanishing elements
本文利用消失元阶集合刻画李型单群 $E_8(p)$，提出验证其 V-可识别性的方法，并在 $p<10000$ 的 610 个相关素数中确认除 7 个例外外均有效。
We calculate the precise number of r-regular elements in the finite exceptional groups. As a corollary we find that the proportion of r-regular elements is at least 3577/18432 and for all ε>0, there are infinitely finite simple exceptional groups such that the proportion of r-regular elements is less than $3577/18432+ε$ for some prime r.
In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.
We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a finite set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n<2n$.
Vanishing Elements of Prime Power Order
本文证明了除 $\mathrm{SL}_2(4)$ 和 $\mathrm{SL}_2(8)$ 外的非交换有限单群均含共轭类大小被三个不同素数整除的素数幂阶消失元素，并据此推广了Robati关于非可解有限群的相关结论。
A Proof of the Isaacs_Navarro_Wolf Conjecture
本文证明了有限可解群的非零元必属于Fitting子群，通过结合Isaacs等人的归约与有限辛空间、Clifford理论完成了Isaacs_Navarro_Wolf猜想的证明。
We give a precise formula and a simple asymptotic function for the proportion of elements with only negative (or only positive) cycles in Weyl groups of type $B,C$ and $D$.
An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural properties of a factorised group $G=AB$ when we impose some conditions on prime power order elements $g\in A\cup B$ which are (non-)vanishing in $G$.
In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the "correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has been made in a number of different directions, most notably by Braue…
We explore the Mayer-Vietoris sequence developed by Chiswell for the fundamental group of a graph of groups when vertex groups satisfy some vanishing assumption on the first cohomology (e.g. property (T), or vanishing of the first $\ell^2$-Betti number). We characterize the vanishing of first reduced cohomology of unitary representations when vertex stabilizer have property (T). We find necessary and sufficient conditions for the vanishing of the first $\ell^2$-Betti number. We also study the associated Haagerup cocycle and show that it vanishes in first reduced cohomology precisely when the a…
In this paper, we get an elementary and important lemma(See Lemma 3.2) which is about pushout and pullback of modules. And we prove a weak form of a long open conjecture on vanishing of group cohomology for blocks.
Let $G({\mathbb F}_{q})$ be a finite Chevalley group defined over the field of $q=p^{r}$ elements, and $k$ be an algebraically closed field of characteristic $p>0$. A fundamental open and elusive problem has been the computation of the cohomology ring $\opH^{\bullet}(G({\mathbb F}_{q}),k)$. In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type $A_n$ and $C_n$, the first non-trivial cohomology classes are determined when $p$ is larger than the Coxeter number (larger than twice the Coxeter number for type $A_n$ with $n>1$ and $r >1$). In …
Vanishing elements of prime power order and their class size property
该研究在Dolfi和Lucido的工作基础上引入了群的性质$P_v(p,q)$，证明有限单群不满足该性质，并推广了相关可解性结果。
The proportion of derangements characterizes the symmetric and alternating groups
本文证明了对称群及其子群中无不动点元素的比例能刻画对称群与交错群，并给出了在monodromy groups中的应用。
We show that it is impossible to algorithmically decide if the l^2-cohomology of the universal cover of a finite CW complex is trivial, even if we only consider complexes whose fundamental group is equal to the elementary amenable group (Z_2 \wr Z)^3. A corollary of the proof is that there is no algorithm which decides if an element of the integral group ring of the group (\Z_2 \wr Z)^4 is a zero-divisor. On the other hand, we show, assuming some standard conjectures, that such an algorithm exists for the integral group ring of any group with a decidable word problem and a bound on the sizes o…
Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\ $when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $% E_{n}(R)$ the group generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained for $\mathrm{Sp}_{2n}(R)$ the symplectic group, $\mathrm{ESp}_{2n}(R)$ the elementary symplectic group, $\mathrm{O}(n,n)(R)$ the split orthogonal group or $\mathrm{EO}(n,n)(R)$ the …
On common zeros of characters of finite groups
本文研究有限群的零化元素与零化共轭类问题，重点探讨了使所有非线性不可约复特征标均在其上取值为零的最小共轭类数量，以及存在公共零点的非线性不可约特征标的最小数量。
Proportion of chiral maps with automorphism group $\mathcal{S}_n$ and $\mathcal{A}_n$
研究证明，当n趋近于无穷大时，具有自同构群S_n或A_n的可定向正则地图中镜像地图的比例趋近于1，并给出了相关渐进行为。
Derangements in finite classical groups and characteristic polynomials of random matrices
研究有限经典群中具有特定特征多项式的元素比例上限，证明高维自然模下元素落在适当不可约子群中的比例趋近于零，并探讨SL(n,q)中三个随机元素生成的概率趋近于零。
For any discrete, torsion-free subgroup $Γ$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(Γ;V)=0$ (resp.\ $H_i(Γ;V)=0$ for $i=13,14,15$) for any $Γ$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $Γ$, improving on work of M.~Kapovich.
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