V. S. Monakhov, A. A. Trofimuk
We consider the structure of a finite groups having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.
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In this paper, we classify the finite simple groups with an abelian Sylow subgroup.
We determine the ranks of the Sylow 2-subgroups of the classical simple groups of odd characteristic.
We determine the ranks and normal ranks of the Sylow 2-subgroups of the classical groups of odd characteristic.
A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup of $G$ is submodular in $G$, then $G$ is called strongly supersoluble. The properties of groups with submodular Sylow subgroups are obtained. In particular, we proved that in a group every Sylow subgroup is submodular if and only if the group is Ore dispersive and every its biprimary subgroup is strongly supersoluble.
The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
Let $G$ be a finite solvable group. We show that $G$ does not have a normal nonabelian Sylow $p$-subgroup when its prime character degree graph $Δ(G)$ satisfies a technical hypothesis.
A subgroup $A$ of a group~$G$ is said to be {\sl NS-supplemented} in $G$, if there exists a subgroup~$B$ of $G$ such that $G=AB$ and whenever $X$~is a normal subgroup of~$A$ and $p\in π(B)$, there exists a Sylow $p$-subgroup~$B_p$ of~$B$ such that $XB_p=B_pX$. In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
The Sylow p-subgroups of the symmetric group S_p^n satisfy the appropriate generalization of Maschke's Theorem to the case of a p'-group acting on a (not necessarily abelian) p-group. Moreover, some known results about the Sylow p-subgroups of S_p^n are stated in a form that is true for all primes p.
An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characterictic.
A note on conjugacy problem for finite Sylow subgroups of infinite linear groups
本文证明了在存在至少一个有限Sylow 2-子群的条件下，伪有限$\mathfrak{M}_c$群中Sylow 2-子群的共轭性，并通过Platonov的反例说明伪有限性假设不可或缺。
The Sylow $p$-subgroups of the finite simple groups
本文总结了经典Lie型有限单群的Sylow $p$-子群。
Normalizers of Sylow subgroups in finite reflection groups
本文证明了有限反射群中西罗子群正规化子的半直积分解，并利用抛物子群与考克斯特图自同构给出实型情形的具体描述。
We determine, for $p$ odd, all saturated fusion systems on a Sylow $p$-subgroup $S$ of the unitary group $SU_4(p)$ and we prove that they are all realizable by finite groups. In particular, we prove that $S$ does not support any exotic fusion systems.
Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this result with recent work of Breuer and Guralnick, we deduce that if $G$ is a finite nonabelian simple group and $r$ is any prime divisor of $|G|$, then $G$ is generated by a Sylow $2$-subgroup and a Sylow $r$-subgroup.
If $G$ is an omega-stable group with a normal definable subgroup $H$, then the Sylow-$2$-subgroups of $G/H$ are the images of the Sylow-$2$-subgroups of $G$.
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$ with $n\in\mathbb{N}$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
The powerful class of Sylow subgroups of finite groups
本文研究有限群中 Sylow p-子群的 powerful 类如何影响转移与融合控制，并给出 p-可解群 p-长度关于该 powerful 类的显式上界。
Let $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $σ(G)=\{σ_{i}|σ_{i}\cap π(|G|)\neq\emptyset\}$. A subgroup $S$ of a group $G$ is called a $σ_i$-sylowizer of a $σ_i$-subgroup $R$ in $G$ if $S$ is maximal in $G$ with respect to having $R$ as its Hall $σ_i$-subgroup. The main aim of this paper is to investigate the influence of $σ_i$-sylowizers on the structure of finite groups. We obtained some new characterizations of supersoluble groups by the permutability of the $σ_i$-sylowizers of some $σ_i$-subgroups.
On Oliver's $p$-group conjecture for Sylow subgroups of unitary groups
该研究针对 Oliver 的 $p$-群猜想，采用初等方法证明其对酉群的 Sylow $p$-子群成立。
Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most $d+η$ with $η\sim 2.875065.$
Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions
本文解决Brauer问题33，确定具有单个对合共轭类的有限群的西洛2-子群，除表1中非阿贝尔单群对应的子群外，还包括同循环群等，用到有限单群分类。
Fusion Systems and Simple Groups With Class Two Sylow $p$-subgroups
本文确定了支撑在幂零类二有限 $p$-群上的所有约化饱和融合系统，并由此给出了具有类二 Sylow $2$-子群的有限简单群分类的新证明。
In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of R.~Skuratovskii "Corepresentation of a Sylow p-subgroup of a group S_n" \cite{Dm, Sk, Paw} is continued. Let $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ be Sylow 2-subgroups of corresponding alternating groups $A_{2^k}$ and $A_{n}$. We find a least generating set and a structure for such subgroups $Sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $Syl_2{A_{n}}$ and commutator width of $Syl_2{…
Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|…
Random unipotent Sylow subgroups of groups of Lie type of bounded rank
该研究针对有界秩有限单李型群，证明其可表为11个随机幂零Sylow子群的乘积且概率随群阶趋于无穷趋近1，给出有限线性群应用，证明不依赖有限单群分类。
We say that a finite group $G$ acting on a set $Ω$ has Property $(*)_p$ for a prime $p$ if $P_ω$ is a Sylow $p$-subgroup of $G_ω$ for all $ω\inΩ$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy…
Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…
For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.
Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- I
本文研究菲舍尔和莫诺斯特奇奇异性群中3-子群的无核融合系统分类，证明3-群不支持奇异融合系统，为后续奇素数p的奇异性群Sylow子群融合系统分类奠定基础。
The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group, finding structure these subgroups. The aim of this paper is to research the structure of Sylow 2-subgroups and to construct a minimal generating system for such subgroups. In other words, the problem is not simply in the proof of existence of a generating set with elements for Sylow …
Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p; then G contains a simple group as a subquotient which exhibits the same property. In addition we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p. This provides a solution to the problem which was left remaining after Tiep and Navarro established that is p is not equal to 3 or 5, then this can never happen [NT14].
Intersections of Sylow 2-subgroups in symmetric groups
本研究计算对称群$S_n$与交错群$A_n$中随机Sylow 2-子群平凡相交的渐近概率，补充了相关学者的近期研究成果。
Sylow subgroups for distinct primes and intersection of nilpotent subgroups
研究不同素数的Sylow子群与nilpotent子群交集的性质，提出并验证了关于包含最小性的猜想，并在特定群类中得出了结论。
One part of Sylow's famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer $n\equiv 1\pmod{p}$ occurs as the number of Sylow p-subgroups of some finite group. While Hall's proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.
Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups
本研究针对几乎单群，证明其特征标表可算法判定Sylow 3-子群是否2生成，且该性质等价于主块特征标的Galois作用条件，是Alperin-McKay-Navarro猜想的推论。
Define a random variable $ξ_n$ by choosing a conjugacy class $C$ of the Sylow $p$-subgroup of $S_{p^n}$ by random, and let $ξ_n$ be the logarithm of the order of an element in $C$. We show that $ξ_n$ has bounded variance and mean order $\frac{\log n}{\log p}+O(1)$, which differs significantly from the average order of elements chosen with equal probability.
The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group, finding structure of these subgroups. The authors of article "Structure of 2-sylow subgroup of symmetric and alternating group" U.~Dmitruk, V.~Suschansky didn't proof minimality of finding by them system of generators for such Sylow 2-subgroups of $A_n$ and structure of it were found…
Denote by $ν_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ν_p(H)\leqν_p(G)$ for $H\leq G$, however $ν_p(H)$ does not divide $ν_p(G)$ in general. In this paper we reduce the question whether $ν_p(H)$ divides $ν_p(G)$ for every $H\leq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {\ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.
Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most d elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and of index bounded by a function of d and p. This result will be used to prove that the index of the Frattini subgroup of G is bounded in terms of d and p. Upper bounds will be given explicitly for soluble groups.
On the maximal overgroups of Sylow subgroups of finite groups
该研究确定了Sylow r-子群含于唯一极大子群的有限群，通过约化到殆单群完成证明，推广了Aschbacher关于r=2的早期结果，并给出弱次正规子群新结果以研究Baer-Suzuki定理变体。
We define a local Sylow subgroup of a totally disconnected, locally compact group G to be a maximal pro-p subgroup of an open compact subgroup of G. We use these subgroups to define the p-localisation of G, a locally virtually pro-p group which maps continuously and injectively to G with dense image, and describe the relationship between the scale and modular functions of G and those of its p-localisation. In the case of locally virtually prosoluble groups, we consider all primes simultaneously using local Sylow bases.
On redundant Sylow subgroups
本文推广 Maróti–Martínez–Moretó 的定理，证明每个非循环 p-群都可作为某个可解群中的冗余 Sylow p-子群，并回答相关公开问题。
A note on blocks of finite groups with TI Sylow $p$-subgroups
本文针对特征零代数闭域上的有限群块，构造了一个函子等价但不完全等距的具体例子，从而说明函子等价并不蕴含完全等距。
Splitting the center of a Sylow subgroup
本文研究了Sylow子群中心的分解问题，证明了在某些条件下Sylow子群中心的结构，并扩展到融合系统。
Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- II
该研究对散在群$\mathrm{Fi}_{24}'$和$\mathrm{M}$的西罗3-子群上的无核融合系统进行分类，完成了奇数$p$时散在群西罗$p$-子群无核融合系统分类项目，还扩展理论并描述算法，大幅加快找$p$-群无核融合系统的过程。
Bicyclic biskew braces
研究加法和乘法群均为循环群的有限支架，从置换群正则子群角度重新分类，并识别其中哪些是双斜支架。
A study of the Wamsley group and its Sylow subgroups
本文研究一类由三个生成元定义的 Wamsley 群及其 Sylow 子群，在参数满足特定非平凡条件下分析其结构，并在 α=β 时取得最精确的结果。
Sylow subgroups of the Macdonald group on 2 parameters
本文研究2参数Macdonald群 $G(α,β)$，填补了其幂零性原始证明中的漏洞，并确定了该群所有Sylow子群的阶与幂零类。
Let $π$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_π\mathfrak{F}$ of all groups $G$, such that $π(G)\subseteq π(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\inπ\capπ(G)$ are investigated. It is established that ${\rm w}^{*}_π\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_π\mathfrak{F}=\mathfrak{F}$ are founded.
The saturated fusion systems on a Sylow 2-subgroup of $Ω^+_8 (2)$
研究$Ω^+_8(2)$的Sylow 2-子群上的饱和融合系统，其中$O_2(\mathcal F)=1$，探讨其结构与性质，包括例子分析。
Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovk…
Commuting probability for the Sylow subgroups of a profinite group
该研究探讨投射有限群的子群交换概率，证明若投射有限群的Sylow 2-子群与Sylow 3、5-子群的交换概率为正则群是虚投射可解的，还给出投射可解群为虚投射幂零的概率判据。
Commuting probability for the Sylow subgroups of a finite group
本文研究有限群中Sylow子群间的交换概率，证明在交换概率有下界时，Fitting子群具有有界指数，并给出幂零性与可解性的阈值判据。
Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+κ$ with $κ\sim 2.75239495.$ The number $κ$ is explicitly described in terms of the Riemann zeta function and is best possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+κ^*$ with $κ^* \sim 1.606695.$
This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of cyclic groups we investigate its commutator width and some properties of its commutator subgroup. It was proven that the commutator width of an arbitrary element of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. As a corollary, it is shown that the commutator width of Sylows $p$-subgroups of symmetric and alternating groups $p…
A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases Cayley graphs of $P_n(2)$ may have different girths (i.e. minimal lengths of cycles) and thus be non-isomorphic. In presented paper all possible values of girths of Cayley graphs of $P_n(2)$ on diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented. A lower bound for the number of different non-isomorphic…
On finite groups isospectral to groups with abelian Sylow $2$-subgroups
研究了与非阿贝尔单群与阿贝尔2-子群的直积具有相同谱的有限群，证明了存在无限多个具有相同谱的有限群。
Simple Skew Braces with Cyclic Sylow Subgroups
研究有限斜括号在西罗子群循环性假设下的单性与分裂现象，先分类乘法群为\(Z\)-群的有限单斜括号，再考虑更一般情况，证明相关结论并验证拜奥特可解性猜想。
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