Edith Adan-Bante
Let $G$ be a finite nilpotent group, $χ$ and $ψ$ be irreducible complex characters of $G$ of prime degree. Assume that $χ(1)=p$. Then either the product $χψ$ is a multiple of an irreducible character or $χψ$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.
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We present a description of non-solvable groups in which all real irreducible character degrees are prime-power numbers.
A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments, nor representation theory.
We construct solvable groups where the only degree of an irreducible character that is a prime power is $1$ and that have arbitrarily large Fitting heights. We will show that we can construct such groups that also have a Sylow tower. We also will show that we can construct such groups using only three primes.
We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.
Minimal degrees for faithful permutation representations of groups of order $p^6$ where $p$ is an odd prime
本文确定了奇素数 $p$ 下所有 $p^6$ 阶群的忠实置换表示的最小次数，并给出了构造方法。
In the process of computing the Galois group of a prime degree polynomial $f(x)$ over $\mathbb Q$ we suggest a preliminary checking for the existence of non-real roots. If $f(x)$ has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of $f(x)$ contains $A_p$ or is one of a short list. Let $f(x)\in \mathbb Q [x] $ be an irreducible polynomial of prime degree $p \geq 5$ and $r=2s$ be the number of non-real roots of $f(x)$. We show that if $s$ satisfies $s (s \log s + 2 \log s + 3)…
In this paper we investigate families of connected graphs which do not contain an odd cycle in their complement. Specifically, we consider graphs formed by two complete graphs connected in a particular way. We determine which of these graphs can or cannot occur as the prime character degree graph of a solvable group. An obvious expansion and generalization can also be considered, of which we make mention.
On prime character degree graphs occurring within a family of graphs (iii)
通过扩展构造的灵活性，完整分类了可解群的素数特征度数图中可能出现的图。
Characters of prime power degree in principal blocks
本文研究并描述了主块中仅包含素数幂次方阶特征的有限群。
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},μ)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the G…
We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.
In this paper we present the set of saturated numerical semigroups with prime multiplicity. We also characterize the catenary degree of these semigroups that we acquire. The catenary degree of a numerical semigroup is the variant which measures the distance between factorizations of elements within that numerical semigroup.
The Hurwitz existence problem and the prime-degree conjecture: A computational perspective
通过计算方法研究Hurwitz存在性问题，验证素数度猜想，并提出稳定内存使用的软件架构以进一步探索异常情况。
Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of $\mathrm{GL}(p,\mathbb{C})$-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For $p\leq 3$, w…
Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to ${\rm PSL}_2 (q)$ for $q \geq 7$.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\rm PSL}_n(q)$, ${\rm PSU}_n(q)$ and ${\rm PSp}_{2n…
Sylow subgroups and the number of irreducible characters of degrees divisible by a prime $p$
本文研究了有限群中 Sylow p-子群的导出长度与可约特征标度数 divisible 由素数 p 的关系，证明了 p-块中缺陷群的导出长度与正高度特征标的数量之间的上界。
In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow $p$-subgroup of $G$ is at most $p$, then $| G:O_p (G) |_p \le b(G)$.
In this paper, we continue the classification work done in the first paper of the same name. With careful modifications of our previous approach, we are able to deduce (with two notable exceptions) which members of the previously introduced graph family manifest as the prime character degree graph of some solvable group.
Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of a…
Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type
研究了有限简单李型群中素数度不可约表示的阶数和数量限制，及其在强逼近定理中的应用。
On prime divisors of character degrees and codegrees
本文研究有限群不可约复特征标的次数和余次数的素因子集合，证明若该集合大小不超过3则群可解，并给出正号情形下的推广。
Non-solvable groups whose non-linear character degrees have the same number of different prime divisors
本文研究非可解群的非线性特征度数具有相同不同素因子数的性质，证明此类群在可解因子作用下仅限于特定几类群，并验证Huppert的ρ-σ猜想。
We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that there are examples that are not $p$-nilpotent where this bound is met for every prime $p$. We then prove a number of variations of this result.
The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there exists an element of order $pq$ in $G$. Let $p_1<p_2<...<p_k$ be all prime divisors of $|G|$. Then the degree pattern of $G$ is defined as ${\rm D}(G)=(deg_G(p_1), deg_G(p_2),..., deg_G(p_k))$, where $deg_G(p)$ signifies the degree of the vertex $p$ in ${\rm GK}(G)$. A finite group $H$ is said to be OD-characterizable if $G\cong H$ for every finite group $G$…
Supersolvability and nilpotency in terms of the commuting probability and the average character degree
针对最小素因子为 $p$ 的有限群，本文给出交换概率和平均特征次数依赖于 $p$ 的尖锐临界界，以判定其幂零性或超可解性。
Given a Mersenne prime $q$ and a positive even integer $e$, let $F$ and $E$ be the fields of orders $q$ and $q^e$ respectively. Let $C$ be a cyclic subgroup of $E^\times$ whose index in $E^\times$ is divisible only by primes dividing $q - 1$. We compute the character degrees of the group $C \rtimes {\rm Gal} (E/F)$ by using the Galois connection between the subfields of $E$ and the Galois group ${\rm Gal} (E/F)$.
$M$-groups and Codegrees; $M_{p}$-groups and Brauer Character Degrees
本文研究有限群的余次数与Brauer特征标次数，证明了具有三个余次数的有限群必为$M$-群，并证明当所有不可约Brauer特征标次数均为素数时，对任意正规子群$N$，$G/N$或$N$中至少有一个是$M_p$-群。
Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for infinitely many groups $G$.
We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley grap…
Let $G$ be a finite group and let $π$ be a set of primes. Write $\mathrm{Irr}_{π'}(G)$ for the set of irreducible characters of degree not divisible by any prime in $π$. We show that if $π$ contains at most two prime numbers and the only element in $\mathrm{Irr}_{π'}(G)$ is the principal character, then $G=1$.
Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $χ\in \tex…
Let $\hat\Z_p$ be the ring of $p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $\hat \Z_p$-modules of degree at most $p$ is equivalent to the category of finitely generated modules over a particularly well understood ring, called Green order. That this is the case was conjectured by Yuri Drozd.
We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that depends on the prime $p$. If we further restrict our average by fixing a subfield $k$ of the complex numbers and then compute the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then we will see that we obtain a bound that depends on both $p$ and $k$. Moreover, we find examples that make tho…
A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $ω\in Ω$ such that $ω^g = ω^h$. The \emph{intersection density} $ρ(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_ω|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_ω$ is the stabilizer of $ω$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $ρ(G) = 1$. Moreover, if $G$ is primitive of de…
Permutation groups of prime power degree and $p$-complements
本文分类了具有素数幂次传递置换表示的几乎单群及其$p$-补，证明本原群有正则子群且同幂次忠实表示自同构等价，并构造了补不等价类任意多的例子。
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。
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the o…
The classical Itô-Michler theorem states that the degree of every ordinary irreducible character of a finite group $G$ is coprime to a prime $p$ if and only if the Sylow $p$-subgroups of $G$ are abelian and normal. In an earlier paper, we used the notion of average character degree to prove an improvement of this theorem for the prime $p=2$. In this follow-up paper, we obtain a full improvement for all primes.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.
Functional degrees and arithmetic applications III: Beyond Prime Exponent
本文将素数 Ax-Katz 定理推广到任意有限交换群之间的多个映射，用不变因子和函数次数给出联立零点集基数的 p-adic 可除性下界。
Primes and The Field of Values of Characters
本文扩展了关于素数p的特征标字段分类推测，适用于特征标次数可被任意次幂p整除的情况，并提供支持其有效性的证据。
Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by ρ(G) the set of primes which divide some character degrees of G. The prime graph Δ(G) associated to G is a graph whose vertex set is ρ(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. In this paper, we show that the prime graph Δ(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices.
Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those powers of $x$ that occur as terms of $L$ with nonzero coefficient have exponent a power of $q$. If the exponent of the leading term of $L$ is $q^n$, we say that $L$ has $q$-degree $n$. We assume that the coefficient of the $x$ term of $L$ is nonzero. We investigate the Galois group $G$, say, of $L$ over $F$, under the assumption that $L(x)/x$ is irreducible …
Functional degrees and arithmetic applications II: The Group-Theoretic Prime Ax-Katz Theorem
本研究将Ax-Katz p进同余式与Moreno-Moreno p权细化推广到任意素特征有限交换环，通过纯群论结果推导零点个数的p进可除性下界，结合Wilson的证明方法与Aichinger-Moosbauer函数演算完成论证。
We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs $Γ_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs $Γ_{k,t}$ occur as the prime character degree graph of a solvable group.
Hall $π$-subgroups and characters of $π'$-degree
本文探讨了有限群中Hall π-子群与π'度数特征标之间的关系，并将Navarro和Tiep的结果推广到多个奇素数的情况。
Classifying Prime Character Degree Graphs With Eight Vertices
本文致力于分类具有八个顶点的素数特征度图，通过现有结果和构造开发通用算法，对八顶点图进行分类，发现37个连通图出现，56个图不出现，206个图尚未分类。
Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let ρ(G) be the set of all primes which divide some character degree of G. The prime graph Δ(G) attached to G is a graph whose vertex set is ρ(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph Δ(G) has no triangles, then Δ(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occ…
The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $δ(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$, see [18]. For $r\geq 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $δ(\mathcal{P}(C_n))$ is equal to the …
Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this pap…
Hopf--Galois structures of cyclic type on parallel extensions of prime power degree
本文研究了当扩展L/K的度数为素数幂且类型为循环时，平行扩展是否也具有相同Hopf-Galois结构的问题，采用群论方法并结合Greither-Pareigis和Byott的工作予以解决。
Character degrees in principal blocks for distinct primes
本文证明阶被不同素数整除的有限群在特定条件下存在同时位于主块且次数不被素数整除的非平凡不可约特征标，推广了前人结果并完成相关猜想的证明。
We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime~$2$, hence for characters of odd degree.
We verify that every alternating group of degree at most one quadrillion is invariably generated by an element of prime order together with an element of prime power order.
We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for som…
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