On the orders of the non-Frattini elements of a finite group
Andrea Lucchini
设$G$是一个有限群，$p_1,\dots,p_n$是不同的素数.如果$G$包含阶为$p_1\cdots p_n,$的元素，则$G$中存在不包含在$G$的弗拉蒂尼子群中且其阶可被$p_1\cdots p_n.$整除的元素
Let $G$ be a finite group and let $p_1,\dots,p_n$ be distinct primes. If $G$ contains an element of order $p_1\cdots p_n,$ then there is an element in $G$ which is not contained in the Frattini subgroup of $G$ and whose order is divisible by $p_1\cdots p_n.$
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The Frattini subgroup for subgroups of hyperbolic groups
本文证明了：对于字双曲群$H$的一个生成子群$H$，$H$的弗拉蒂尼子群$F(H)$是有限的.
Recognizing $\mathrm{PSL}(2,p)$ in the non-Frattini chief factors of finite groups
给定一个有限群$G$，令$P_G(s)$是$s$随机选择的元素生成$G$的概率，令$H$是具有$P_G(s)=P_H(s)$的有限群。证明了如果对某些非Mersense素数$p\geq 5$，$G$和$H$的非交换合成因子为$\mathrm{PSL}(2,p)$，则$G$和$H$具有相同的非Frattini主因子.
On the Frattini subgroup of a finite group
本文研究了有限群类$G$满足$Φ(G/N)= Φ(G)N/N$，对$G$的所有正规子群$N$.由于我们的主要结果，我们延长和扩大定理的Doerk关于这个类从可溶性宇宙的所有有限群和答案肯定的一个长期存在的问题克里斯滕森是否类有限群拥有补充其每个正规子群是次正规封闭的。
On the Frattini lemma
设$K$是有限群$G$的一个子群，对$K$的每个Sylow子群$P$，有$G=KN_G(P)$.那么亚组$K$在$G$中是正常的。
Frattini Argument for Hall subgroups
本文证明了：若有限群$G$对素数集$π$有一个$π$ -Hall子群，则$G$的每个正规子群$A$有一个$π$ -Hall子群$H$，使得${G=AN_G(H)}$ .
On the Cohomology of Central Frattini Extensions
利用拓扑方法计算了某些p-群的模p上同调.更确切地说，我们期待在中央弗拉蒂尼扩展的基本阿贝尔的基本阿贝尔群，使其定义的k-不变量跨越整个图像的博克斯坦。我们表明，如果p是足够大的，那么模p上同调的扩展可以明确计算为一个代数。
We call a pro-$p$ group $G$ Frattini-injective if distinct finitely generated subgroups of $G$ have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-$p$ groups (and several other related concepts). Most notably, we classify the $p$-adic analytic and the solvable Frattini-injective pro-$p$ groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-$p$ group. We prove that every maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is Fr…
Frattini subgroups of hyperbolic-like groups
该研究探讨类双曲群的Frattini子群，证明一般型作用于双曲空间的可数群的Frattini子群诱导作用轨道有界、指数无限，同时构造出Frattini子群指数有限的无限lacunary hyperbolic群，并给出首批不变生成的此类无限群实例。
On Frattini arguments in L-groups of finite Morley rank
我们修改了卡特子群的Frecon-Jaligot构造，证明了退化型群在自同构群的Sylow 2-子群下有一个卡特子群不变量，从而减少了需要知道退化型群中卡特子群是共轭的.
Forcible groups and Frattini covers
本文补充Thompson-Mann经典结果并回答相关问题，证明有限群可强制的充要条件，同时给出阿贝尔p-群在两类p-群下的相对可强制性结果。
Lifting in Frattini Covers and A Characterization of Finite Solvable Groups
证明了有限群的奇Frattini覆盖的一个提升定理。利用这一点，我们的特点可解群和更一般的p-可解群在包含一个三元组的元素不同的素数幂次与产品1。这也涉及到各种问题弗里德的模块塔计划和性质的Hurwitz空间覆盖的曲线。
On the local Frattini subgroup of a finite group
本研究针对有限群的局部Frattini子群，在已有学者相关结论基础上，补充了更多能保证该子群包含于Frattini子群的条件。
Maximal linear groups induced on the Frattini quotient of a $p$-group
设$p>3$为素数。对于每个具有$|H| \geqslant p^{3d+1}$的极大子群$H\leqslant\mathrm{GL}(d,p)$，我们构造了一个$d$生成元有限的$p$群$G$，其性质是$\mathrm{Aut}(G)$在Frattini商$G/Φ(G)$和$|G| \leqslant p^{\frac{d^4}{2}}$上诱导$H$.这种结构的一个显著特点是，与$|H|$相比，$|G|$非常小，这为布莱恩特和科瓦奇的一个著名结果提供了新的线索。我们展示的群$G$具有指数$p$，并且在所有具有$H$对$G/Φ(G)$的期望作用的这样的群$G$中，构造产生具有最小幂零类的群，并且在大多数情况下，产生最小阶的群。
Finite 2-groups of Class 2 with Specific Automorphism Group
本文对所有2类有限2-群进行了分类，其中每个2阶自同构离开弗拉蒂尼子群元素不动都是内的.我们证明了每个这样的群G同构于Q（n; r）= <a，b| a^{2n}= b^{2 r}= 1; a^2^{n-r}= [a，b]>，对于某些正整数r; n，使得2 <2 r <= n;且Q（n; r）的每一个2阶自同构使得弗拉蒂尼子群按元素不动都是内自同构.
Frattini and related subgroups of Mapping Class Groups
设$Γ_{g,b}$表示亏格为$g$的闭可定向曲面的保向映射类群，其中该曲面具有$b$穿孔.对于群$G$，令$Φ_f(G)$表示$G$中所有有限指数极大子群的交。基于Ivanov提出的当$G$是$Γ_{g,b}$的一个生成子群时$Φ_f(G)$是否为幂零的问题，本文计算了$Γ_{g,b}$的某些子群的$Φ_f(G)$.特别是，我们回答伊万诺夫的问题是肯定的，这些子群的$Γ_{g,b}$。
A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto Φ(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois theory. Indeed, every maximal pro-$p$ Galois group over a field that contains a primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ is torsion-fre…
Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups
该研究针对定向右角Artin pro-$p$群，证明了其关联有向图为初等型、具有Frattini抗性、拓扑有限生成闭子群均为定向pro-$p$ RAAG、是含$p$次单位根域的极大pro-$p$伽罗瓦群这四个条件等价，并提出相关上同调无本质三重Massey积的猜想。
On finite capable $p$-groups of class 2 with cyclic commutator subgroups
研究幂零类2的有限能力$p$ -群$G$，使得$G$的换位子群$γ_2(G)$是循环的，且$G$的中心包含在$G$的弗拉蒂尼子群中.
Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers
本文通过Frattini拓扑约化将有限群的非生成复形化为初等阿贝尔向量空间的非张成复形，从而确定其同构、自同构群、同调及Betti数，并给出完整Cohen-Macaulay分类。
Finite $p$-groups of class 2 have noninner automorphisms of order $p$
我们证明了对任何素数$p$，每个类2的有限非交换$p$ -群$G$都有一个阶为$p$的非内自同构，使弗拉蒂尼子群$Φ(G)$或$Ω_1(Z(G))$元素固定.
It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.
An Application of Maximality to Nilpotent and Finitely Generated L-Subgroups of an L-Group
本文继续研究 L-群的极大和 Frattini L-子群，探讨了幂零 L-群中极大 L-子群的正规性，并引入有限生成 L-子群概念，建立其与极大条件的关系及相关性质。
For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
Groups with a base property analogous to that of vector spaces
B-群是一个群，使得它的所有极小生成集（关于包含）具有相同的大小。我们证明了有限B-群类在取同元下是闭的，并且每个有限B-群都是可解的。通过对Frattini-free有限B-群的完全分类，得到了有限B-群的一般结构定理。其应用包括有限拟阵群的刻画和具有基性质的有限群的分类的新证明。
On finite groups whose Sylow subgroups have a bounded number of generators
设G是有限非幂零群，使得G的每个Sylow子群至多由d个元生成，且p是除|G|的最大素数.我们证明了G有一个非幂零像G/N，使得N是特征的，且其指数由一个函数d和p有界.这个结果将被用来证明G的弗拉蒂尼子群的指数是有界的.对于可解群，我们将给出明确的上界.
Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2
本文提出首个对Frattini类2的2-群在$N^{O((\log N)^{1/2})}$时间内进行同构测试的算法，利用二次型等距问题和矩阵群算法，突破了$p=2$情况下的长期瓶颈。
Countable Primitive Groups
本文给出了可数本原群在线性群、映射类群的子群、极小作用于树的群和收敛群等情形下的完全刻划。后一类包括作为特殊情况克莱因群以及字双曲群的子群。作为一个应用程序，我们计算的弗拉蒂尼子群在许多这些设置，往往概括的结果，只知道有限生成群。特别地，我们回答了G. Higman和B.H.诺依曼论Frattini群的一个合并积。
Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b(G,H)$ denote the base size for the action of $G$ on $G/H$. Let $α(G)$ be the number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H) \geqslant 3$. We say that $G$ is a strongly base-two group if $α(G) \leqslant 1$, which means that almost every faithful transitive permutation representation of $G$ has base size $2$. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup.
From almost tight $p$-group enumeration to $2$-group domination among finite groups
该研究对有限p群的渐近计数得到几乎紧的界，回答了经典群论问题，证明阶不超过n的有限群中几乎都是幂零类为2的2群，改进并匹配了相关计数的上下界。
A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, Φ(G)\rangle = \langle X, Φ(G)\rangle$, where $Φ(G)$ is the Frattini subgroup of $G$. A group $G$ is $\mathcal{B}_{pp}$ if all prime-power-independent generating sets for $G$ have the same cardinality. We prove that, if $G$ is $\mathcal{B}_{pp}$, then $G$ is solvable. Pivoting on some recent results of Krempa and Stocka, this yields a complete classification of $\mathcal{B}_{pp}$-groups.
On finite $p$-groups with abelian automorphism group
首次构造了具有交换自同构群的各种类型的特殊非特殊有限$p$ -群。更具体地说，我们构造了具有交换自同构群的群$G$，使得$γ_2(G) < \mathrm{Z}(G) < Φ(G)$，其中$γ_2(G)$，$\mathrm{Z}(G)$和$Φ(G)$分别表示$G$的换位子群，中心群和Frattini子群.对于有限的具有初等阿贝尔自同构群的$p$ -群$G$，我们证明了以下两个条件中至少有一个成立：（i）$\mathrm{Z}(G) = Φ(G)$是初等阿贝尔群;（ii）$γ_2(G) = Φ(G)$是初等阿贝尔群，其中$p$是奇素数.我们构造例子来证明具有初等阿贝尔自同构群的群$G$的存在性，对于这类群，上述两个条件之一恰好成立。
Powerfully embedded subgroups of extensions of powerful pro-$p$ groups and its applications to automorphisms of finite groups
本文研究了pro-p群的幂ful子群结构，证明了在特定条件下，这些子群的下p-级数和下主级数项在群中被幂ful地嵌入，并探讨了交叉平方的性质。
On the conjecture of non-inner automorphisms of finite $p$-groups
针对有限非交换 $p$-群必有阶 $p$ 非内自同构的猜想，本文在奇数阶单块群且极大子群满足中心换位条件时，构造并证明了逐元固定 $\Phi(G)$ 的阶 $p$ 非内自同构。
p-groups having a unique proper non-trivial characteristic subgroup
我们考虑有限$p$ -群$G$的结构，它恰好有三个特征子群，即$1$，$Φ(G)$和$G$ . $G$的结构根据$G$是否具有指数$p$或$p^2$而显著变化，并且在这两种情况下，对这些组的研究在表示论中提出了深层次的问题。我们提出了3-和4-生成元群的分类定理，并研究了对于不同的$r$值，指数为$p^2$的$r$ -生成元群的存在性. Frattini商上的自同构群在各种情况下与Aschbacher分类方案中的极大线性群有关。
Pro-$p$ groups of positive rank gradient and Hausdorff dimension
本文研究了正秩梯度有限生成pro-$p$群的Hausdorff维数，推广了Greenberg定理，并证明此类群及其直积关于Frattini级数和Zassenhaus级数具有完整的Hausdorff谱，解决了该领域长期存在的问题。
Group algebras with unit group of class $p$
设V（F_pG）是p元域F_p上有限非交换p-群G的群代数F_pG的正规化单位群。研究了幂零类为p时V（F_pG）的幂结构，证明了若G和H是p-群弗拉蒂尼子群且p>2，则V（F_pG）同构于V（F_pH）当且仅当G和H同构.
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-s…
On complemented non-abelian chief factors of a Lie algebra
由\cite [定理2.3]{[11]}证明了有限维李代数$L$的Frattini主因子或由极大子代数补的主因子的个数在$L$的每个主级数中是相同的。然而，对于在$L$中被简单补的主因子的数目，情况并非如此。在本文中，我们确定了这个数字的可能变化。
The covering number of a nontrivial finite group $G$, denoted $σ(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $ι(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $ι(G)$ when $G$ is a nontrivi…
Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal size of a maximal irredundant set of maximal subgroups of $G$. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that ${\rm Mindim}(G) \leqslant 3$ for all finite simple groups, which is b…
Almost all finite groups are $2$-groups of class two
该研究证明几乎所有有限群是二阶幂零类的2-群，给出阶为p^m的群数量公式，还确定其最小生成元数量分布的渐近特征，改进了p-群数量的经典估计。
Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $α(G)$, which is the minimal number of maximal subgroups whose intersection coincides with ${\rm Frat}(G)$. In earlier work, we studied $α(G)$ in the special case where $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that $α(G) \leqslant 4$ for every almost simple group $G$, which is best possible. We also establish new results on the intersect…
On the cohomology of finite-dimensional nilpotent groups and Lie rings
研究有限维 nilpotent 群和 Lie 环的第一上同调群在不变子模为零时的消失性质，采用模型论方法，发展了基本代数方法，并应用于 Cartan 子环的 Frattini 论证和 definable 连接 p-可除交换群的 Maschke 定理变体。
The structure of a finite group and the maximum $π$-index of its elements
本文研究了有限群元素的极大$π$-指标如何限制商群$G/Z(G)$的结构，证明了其Sylow $p$-子群的Frattini长度及$π$-可分群的$π$-长度均受极大$π$-指标上界控制。
On average orders of automorphism groups of bilinear maps over finite fields
本文研究有限域上双线性映射在一般线性群作用下的同构类数量渐近界，并将其应用于 Frattini 类 2 的 p-群计数、平凡自同构群矩阵空间采样及立方零交换代数同构类估计。
Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups
该研究确定了双循环群等的一致交数，通过基于子群的生成图$\boldsymbol{\textit{G}}(G)$刻画交数为2的群，分类有限群，并证明$\boldsymbol{\textit{G}}(\boldsymbol{\textit{Z}}_n)$为哈密顿图的充要条件。
Automorphism Classes of Elements in Finitely Generated Abelian Groups
我们将证明一个无重复生成的阿贝尔群的每个元素是自同构等价的，我们将定义为{\em repeat-free子群}中的{ \em代表元素}，并且对于有限的阿贝尔群，我们可以计算元素的自同构类的数量。
Virtually free pro-p groups whose torsion elements have finite centralizer
一个扭元具有有限个中心化子的拟自由pro-p群是有限个p-群与一个自由pro-p因子的自由pro-p积.
Element Orders and Character Codegrees of Finite Groups
证明了有限群中元素阶与不可约特征标余次数关系的Qian猜想。
Generation of finite simple groups by an involution and an element of prime order
证明了每个非交换有限单群都是由一个对合和一个素阶元生成的。
Second Countable Virtually Free Pro-p Groups whose Torsion Elements have Finite Centralizer
第二个可数虚自由pro-p群是有限p-群与自由pro-p因子的自由pro-p积，它的所有扭元都有有限中心化子.
The exact number of r-regular elements in finite exceptional groups
我们计算了有限例外群中r-正则元的精确个数。作为推论，我们发现r-正则元的比例至少为3577/18432，并且对于所有ε>0，存在无限有限的单例外群使得r-正则元的比例对于某个素数r小于$3577/18432+ε$.
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.
A bound on element orders in the holomorph of a finite group
设$G$是一个有限群.本文证明了一个定理，它意味着全形$\operatorname{Hol}(G)$的元素的阶由$|G|$从上有界，并讨论了它在有限群的有界自同构阶中的应用.
Locally finite groups containing a $2$-element with Chernikov centralizer
设局部有限群$G$有一个$2$ -元$g$具有Chernikov中心化子.证明了如果$\langle g\rangle$中的对合有幂零中心化子，则$G$有有限指数的可解子群.
Odd-degree rational characters and the order of rational elements in finite groups
本研究证明了有限群中有理不可约特征标均为奇次时所有有理元素都是2-元素的结论，验证了Tiep与Tong-Viet提出的相关猜想。
Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g.$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?
In this note some properties of the sum of element orders of a finite abelian group are studied.
Finite $p$-groups with minimum number of central automorphisms fixing the center element-wise
我们刻画了阶为$p^7$的有限$p$ -群$G$，其中固定中心元素的中心自同构群具有最小可能阶.
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