María-José Felipe, Lev S. Kazarin, Ana Martínez-Pastor, Víctor Sotomayor
Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$.
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We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if $G$ is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime $p$, the $p$-length of such a group is at most $1$. This answers questions proposed by V. Monakhov in The Kourovka Notebook.
Finite solvable tidy Groups whose orders are divisible by two primes
本文通过分类整洁的{p,q}-群，刻画有限整洁可解群，并据此给出其Fitting高的界，同时证明其商群保持整洁性。
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。
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $χ$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$ of $G$ is a finite group of Lie type with rank $n\leq x$. This result is proved using new results connecting the prime graph of $T$ to the integer $x$. We then study the particular cases $x=1$ and $x=2$. We give a general structure statement for $(2,m,n)$-groups which have Euler characteristic a prime power, and we construct an infinite family of these objects. …
Héthelyi and Külshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result is established. It is proved that there exists a constant $c>0$ such that for any finite group $G$ whose order is divisible by the square of a prime $p$ we have $k(G) \geq cp$.
Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the case that all prime power order elements in $A\cup B$ have prime power indices in $G$. In particular, when $G = A = B$ we obtain as a consequence some known results.
Finite groups whose proper subgroups of order divisible by $p$ are supersolvable
研究阶可被奇素数p整除且真子群为超可解但自身非超可解的有限群，通过探讨可解与不可解情形，来分析这类群的算术及结构性质。
Almost all wreath product character values are divisible by given primes
该研究证明，对于任意有限群 $G$ 和素数 $p$，当 $N$ 趋于无穷时，圈积 $G \wr S_N$ 的特征标表中几乎所有条目都能被 $p$ 整除，推广了 Peluse 和 Soundararajan 关于对称群的结果。
On finite groups in which some maximal invariant subgroups have indices a prime or the square of a prime
本文研究有限群在互素自同构群作用下极大不变子群指数为素数或素数平方时的可解性，推广了Hall定理，并完整刻画了非幂零极大不变子群包含某Sylow子群正规化子且指数为素数的有限群。
We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all th…
On soluble groups in which commutators have prime power order
针对换位子阶均为素数幂的有限CPPO群，论文证明可解情形下其换位子子群G'的阶至多含两个素因子。
Finite groups in which every maximal invariant subgroup of order divisible by $p$ is nilpotent
针对有限群$A$互素作用于有限群$G$的情形，对$|G|$的固定素因子$p$，完整刻画了所有阶可被$p$整除的极大$A$-不变子群均为幂零群的$G$的结构。
On the number of conjugacy classes of finite solvable groups of order divisible by $p^2$
该研究修正了有限可解群共轭类数的已有下界，证明原界无法达到，确定了最优界，明确斜率$49/60$最优，加性常数应为$451/60$而非$1/60$。
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。
Indices of non-supersolvable maximal subgroups in finite groups
基于Doerk和Hall关于有限群可解性的经典结果，本文刻画了极大子群为超可解或指数为素数/素数平方的非可解有限群的结构。
Supersolvable subgroups of order divisible by 3
本文确定了所有阶被3整除的极大子群均超可解的有限非可解群的结构，证明其要么是3'-群，要么商群同构于PSL_2(2^p)且满足特定幂零性条件。
Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups
本文为PSL(2,q)-可解群提供素图分类准则，并首次对T属于大（可能无限）群族PSL(2,2^f)的T-可解群证明一般分类结果。
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…
Finite groups in which some particular non-nilpotent maximal invariant subgroups have indices a prime-power
本文研究互素自同构群作用下有限群的结构，证明在特定极大不变子群为同构单群直积、且不含某正规子群的非幂零极大不变子群均具素数幂指数且排除PSL_2(7)作为合成因子的条件下，该群必可解。
The non-two-primes graph of a finite group
本文为任意有限群定义了一种图结构，并证明由其非孤立顶点导出的子图是连通的且直径至多为5。
Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.
The {\it prime graph} $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order $rs$. Let $A_n$ ($S_n$) denote the alternating (symmetric) group of degree $n$. We prove that if $G$ is a finite group with $Γ(G)=Γ(A_n)$ or $Γ(G)=Γ(S_n)$, where $n\geq19$, then there exists a normal subgroup $K$ of $G$ and an integer $t$ such that $A_t\leq G/K\leq S_t$ and $|K|$ is divisible by at most one prime greater than $n/2$.
Finite groups whose maximal subgroups have almost odd index
本文研究了由群不变量的算术性质决定有限群结构的问题，确定了极大子群的指数不被4整除的有限群的结构，并由此得到了一些新的2-幂零性准则。
Let $ρ(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $ρ(n)\leq 5n$ for all $n\geq 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.
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.
Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.
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 ${\rm Gal} (E/F)$.
Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.
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 $G=1$.
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。
The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.
We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to $\operatorname{PSL}(2, p^f)$ for $f \leq 2$, establishing the Prime Graph Question for the first time for all automorphic extensions of series of simple groups. Using this, we determine exactly how far the so-called HeLP-method can take us for (almost simple) groups having an order divisible by at most $4$ different primes.
We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-…
We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if $k$ is a positive integer such that for any prime $p$ the number of character codegrees of a finite group $G$ that are divisible by $p$ is at most $k$, then the number of prime divisors of $|G|$ is bounded in terms of $k$. We prove this conjecture for solvable groups.
On metacyclic p-group codes
本文研究源于有限半单群代数的亚循环p-群码，推广了阶可被任意两素数整除的亚循环群结果，扩展了二面体群和四元数群的现有结论，并利用Bass和双循环单位构造了非中心码。
We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $ρ(g-1)$ for some integer $ρ\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $ρ>6$, and of the first and third authors for $ρ=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; thi…
We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.
Let $G$ be a finite group and, for a prime $p$, let $S$ be a Sylow $p$-subgroup of $G$. A character $χ$ of $G$ is called $\Syl_p$-regular if the restriction of $χ$ to $S$ is the character of the regular representation of $S$. If, in addition, $χ$ vanishes at all elements of order divisible by $p$, $χ$ is said to be Steinberg-like. For every finite simple group $G$ we determine all primes $p$ for which $G$ admits a Steinberg-like character, except for alternating groups in characteristic~2. Moreover, we determine all primes for which $G$ has a projective $FG$-module of dimension $|S|$, where $F…
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