Guohua Qian, Wujie Shi
The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
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We classify finite groups with a small average number of zeros in the character table.
We classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside's classical theorem on zeros of characters.
A group $G$ is called to be acceptable (due to M. Larsen) if for any finite group $H$, two element-conjugate homomorphisms are globally conjugate. We answer the acceptability question for general linear, special linear, unitary, symplectic and all orthogonal groups over an algebraically closed field of non-zero characteristics.
There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros in the character table of $ G $ divided by the number of irreducible characters of $ G $. We show that if $ \mathrm{anz}(G) < 1 $, then the group $ G $ is solvable and also that if $ \mathrm{anz}(G) < \frac{1}{2} $, then $ G $ is supersolvable. We characterise abelian groups by showing that $ \mathrm{anz}(G) < \frac{1}{3} $ if and only if $ G $ is abelian.
Zeros and Roots of Unity for Characters of Solvable Groups
本文证明了Miller猜想：有限可解群的特征标在至少一半元素上取值为零或单位根，并将该界推广到任意有限群的单项不可约特征标。
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 $G$.
$p$-groups and zeros of characters
本文研究了在固定素数 $p$ 和整数 $n$ 的情况下，阶为 $p^n$ 的 $p$-群的非线性不可约特征中取值为 0 的元素的最小数量问题。
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.
Character triples and relative defect zero characters
本文引入了特征标三元组中θ到G的π-拟扩张概念，证明了其在正规化情形下的唯一性，并据此构造了π-亏零特征标与相对π-亏零特征标间的双射，推广了Murai和Navarro的相关定理。
The Field of Values of the Height Zero Characters
本文研究有限群不可约 p-高度零特征标的值域，完整确定 p=2 的情形，并对奇素数提出猜想且归结为拟单群块问题。
For any finite group $G$, Thompson proved that, for each $χ\in {\rm Irr}(G)$, $χ(g)$ is a root of unity or zero for more than a third of the elements $g\in G$, and Gallagher proved that, for each larger than average class $g^G$, $χ(g)$ is a root of unity or zero for more than a third of the irreducible characters $χ\in {\rm Irr}(G)$. We show that in many cases "more than a third" can be replaced by "more than half".
Common zeros of irreducible characters
本文研究有限群不可约特征的公共零点问题，引入公共零点图并证明了对称群特征公共零点存在性，且可解群与单群该图连通分支上界为3，最后应用于证明置换度量的不等价性。
The $p$-rationality of height-zero characters
本文提出并佐证一项全局—局部猜想：有限群中 $p$-高度零特征标的 $p$-有理性可由其亏群的正规化子刻画。
Denseness results for zeros and roots of unity in character tables
本文证明了有限群不可约特征标取值中零点与单位根元素占比在区间[1/2,1]内具有稠密性，即对任意目标比例和误差均可构造出相应的不可约特征标。
Relations between values and zeros of irreducible characters of symmetric groups
本文研究了对称群不可约特征标值与零点之间的关系，证明了某些多项式关系，并改进了通过检查少量值来识别不可约特征标的结果。
Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.
Zeros of $S$-characters
研究$S$-字符的零点性质，发现非平凡$S$-字符在某些群元处取零值，且非平凡传递排列字符在素数幂阶元处消失，但此性质不推广至$S$-字符，回应了Serre的问题。
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is l…
We consider random character values X(g) of the symmetric group on n symbols, where X is chosen at random from the set of irreducible characters and g is chosen at random from the group, and we show that X(g)=0 with probability tending to one as n tends to infinity.
Degrees and prime power order zeros of characters of symmetric and alternating groups
本文研究对称群与交错群不可约特征标的相关性质，证明对称群特征标次数p-部分由p-幂阶消失元决定，交错群p=2时仅存在两种可能，并推广至p-块亏度与特征标p-高度。
For $G={\rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(χ,g)$ in ${\rm Irr}(G)\times G$ with $χ(g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to\infty$.
Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups
本文通过大规模蒙特卡洛模拟逼近大型对称群特征标表中零元的密度，发现与先前结论相反，几乎所有零元均属于单一经典类型。
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