David J. Green, László Héthelyi, Markus Lilienthal
Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain characteristic subgroup X(S) always contains the Thompson subgroup J(S). We obtain a reformulation of the conjecture as a statement about modular representations of p-groups. Using this we verify Oliver's conjecture for groups where S/X(S) has nilpotence class at most two.
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To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.
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。
We introduce a strong form of Oliver's p-group conjecture and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group S_n and of the general linear group GL_n(F_q) satisfy both the strong conjecture and its reformulation.
Let $S$ be a $p$-group for an odd prime $p$, Oliver proposed the conjecture that the Thompson subgroup $J(S)$ is always contained in the Oliver subgroup $\mathfrak{X}(S)$. That means he conjectured that $|J(S)\mathfrak{X}(S):\mathfrak{X}(S)|=1$. Let $\mathfrak{X}_1(S)$ be a subgroup of $S$ such that $\mathfrak{X}_1(S)/\mathfrak{X}(S)$ is the center of $S/\mathfrak{X}(S)$. In this short note, we prove that $J(S)\leq \mathfrak{X}(S)$ if and only if $J(S)\leq \mathfrak{X}_1(S)$. As an easy application, we prove that $|J(S)\mathfrak{X}(S):\mathfrak{X}(S)|\neq p$.
Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\X(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using ideas and work of Glauberman, we prove that if $p \geq 5$, $G$ is a finite $p$-group, and $V$ is an elementary abelian $p$-group which is an F-module for $G$, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in $G$. We apply this to show that Oliver's co…
Let p be an odd prime and S a finite p-group. B. Oliver's conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver's conjecture as a statement about the representation theory of the factor group S/X(S). We now verify the conjecture for a wide variety of groups S/X(S).
This paper is concerned with the Laitinen Conjecture. The conjecture predicts an answer to the Smith question which reads as follows. Is it true that for a finite group acting smoothly on a sphere with exactly two fixed points, the tangent spaces at the fixed points have always isomorphic group module structures defined by differentiation of the action? Using the technique of induction of group representations, we indicate a new infinite family of finite Oliver groups for which the Laitinen Conjecture holds.
We give an alternate computer-free proof of a result of Z. Arad, M. Muzychuk, and A. Oliver: if G is a minimal counterexample to the Sym(3) conjecture, then Soc(G)' cannot be isomorphic to Alt(8).
In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a_G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a_G > 1 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with …
On quadratic conjecture
本文研究极大类 p-群的二次猜想，证明了当阶数参数 n≤8 或 n≥max{2p-6,p+2} 时猜想成立，并由此推出 p≤7 时极大类 p-群均满足二次猜想。
We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete $p$-toral groups with an abelian subgroup of index $p$. In all of these cases (finite or infinite), we reduce the problem to one of listing all $\mathbb{F}_pG$-modules (for $G$ finite) satisfying certain conditions: a problem which was solve…
Twisted Malle's Conjecture
本文运用归纳方法证明了扭曲的Malle猜想的多种新情形，将已有结论从阿贝尔子群等特例推广到更广泛的群结构，拓展了该猜想的适用范围。
In this note, we prove that the $K(π,1)$-conjecture for Artin groups implies the center conjecture for Artin groups. Specifically, every Artin group without a spherical factor that satisfies the $K(π,1)$-conjecture has a trivial center.
We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups.
Kervaire conjecture on weight of group via fundamental group of ribbon sphere-link
本文通过证明纽结外部的猜想Z，证实了Kervaire关于非平凡群与无限循环群自由乘积权重不为1的猜想。
It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.
We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.
The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.
The Picky and Subnormalizer Conjectures for symmetric groups
本文研究了对称群中与麦凯猜想相关的挑拣猜想和次正规化猜想。
In this note we verify the equivalent version of Huppert's conjecture for $K_3$-groups.
We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its *-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.
Harada's conjecture II for the finite general linear groups and unitary groups
本文研究了 Harada 猜想 II 在有限域上一般线性群 $G$ 中的适用性，证明了当域的阶足够大时，所有共轭类大小的乘积可被所有不可约特征标次数的乘积整除。
In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$".
In this note we provide some counterexamples for the conjecture of Moretó on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$. Moreover, we show that this conjecture holds for all sporadic simple groups and alternating groups $A_n$, where $n\neq 8, 10$. Some related conjectures are also discussed.
In this note, I propose the following conjecture: a finite group G is nilpotent if and only if its largest quotient B-group β(G) is nilpotent. I give a proof of this conjecture under the additional assumption that G be solvable. I also show that this conjecture is equivalent to the following: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.
The Codegree Isomorphism Conjecture for Finite Simple Groups
该研究证明有限单群的 codegree 同构猜想，得出非阿贝尔有限单群可由其不可约特征标 codegree 集合在有限群中唯一确定的结论。
We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\inω}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\inω$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.
The Amit-Ashurst conjecture for finite metacyclic p-groups
本研究将广义Amit猜想命名为Amit-Ashurst猜想，并证明该猜想对具有循环极大子群的有限p群成立，推进了有限幂零群字映射概率性质的相关研究。
On Wiegold's conjecture for the small Ree groups
本文证明Wiegold猜想对小Ree群在k≥5的k元组上成立。
Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However it is known to be true for many classes of solvable groups, as for example nilpotent groups, cyclic-by-abelian groups and groups having a cyclic Sylow subgroup with abelian complement. On the other hand, it is not known whether the conjecture holds for supersolvable groups. This paper is a contribution …
The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in [FKMRR]. First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if $G$ is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then $G$ is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results …
In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.
Kropholler and Mislin conjectured that groups acting admissibly on a finite-dimensional G-CW-complex with finite stabilisers admit a finite-dimensional model for E_FG, the classifying space for proper actions. This conjecture is known to hold for groups with bounded torsion. In this note we consider a large class of groups U containing the above and many known examples with unbounded torsion. We show that the conjecture holds for a large subclass of U.
The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews-Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag-Solitar groups do not satisfy the…
The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees $2n$ and $2n-1$ to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for $p+1$ and $p-1$ whenever $p \geq 11$ is a prime number.
On the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions
本文扩展了 Lemke Oliver、Wang 和 Wood 关于 $C_2 \wr H$-扩张类群中 $3$-挠元平均大小有限性的结果，将其从大多数幂零群推广到任意幂零群 $H$，从而更完整地支持了 Cohen-Lenstra-Martinet 启发式猜想。
The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-grou…
We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.
In this paper, we prove Späth's Character Triple Conjecture for $p$-solvable groups. This is a conjecture proposed by Späth during the reduction process of Dade's Projective Conjecture to quasisimple groups. In addition, as suggested by Isaacs and Navarro, we take into account the $p$-residue of characters.
Using the classification of finite simple groups we prove Alperin's weight conjecture and the character theoretic version of Broue's abelian defect group conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.
We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for virtually solvable groups.
We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.
We give an expanded treatment of our lecture series at the 2017 Groups St Andrews conference in Birmingham on local-global conjectures and the block theory of finite reductive groups.
We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
We give an alternate proof of a Theorem of Elek and Szabo establishing Lück's determinant conjecture for sofic groups. Our proof is based on traces on group C*-algebras. We briefly discuss the relation with Atiyah's problem on the integrality of $L^2$-Betti numbers.
Margulis showed that "most" arithmetic groups are superrigid. Platonov conjectured, conversely, that finitely generated linear groups which are superrigid must be of "arithmetic type." We construct counterexamples to Platonov's Conjecture.
We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.
We prove analogues for reductive algebraic groups of some results for finite groups due to Knoerr and Robinson which play a central role in their reformulation of Alperin's conjecture for finite groups.
We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.
The Eaton-Moreto Conjecture and p-Solvable Groups
本文证明了Eaton-Moreto猜想对p-可解群的主块成立，推进了该猜想在有限群表示论中的适用范围。
We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.
We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without employing it.
This paper contains the results of my PhD-thesis. I will show the K- and L-theoretic Farrell-Jones conjecture (FJC) for the general linear groups over the rationals and over the rational functions over a finite field. This especially implies the conjecture for all S-arithmetic groups.
We prove that the alternating groups of degree at least $5$ are uniquely determined up to an abelian direct factor by the degrees of their irreducible complex representations. This confirms Huppert's Conjecture for alternating groups.
We prove that every amenable group of cohomological dimension two whose integral group ring is a domain is solvable and investigate certain homological finiteness properties of groups that satisfy the analytic zero divisor conjecture and act on an acyclic CW-complex with amenable stabilisers.
In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3) that, together with the assumption $|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.
It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I.S. Luthar and I.B.S. Passi which sometimes permits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove with additional arguments that the conjecture is valid for any 3-dimensional crystallographic point group. Finally we apply the method to generic character tables and establish a p-variation of the conjecture …
The Blockwise Navarro Alperin Weight Conjecture for Double Covers of Symmetric and Alternating Groups
本文证明了对称群与交错群双重覆盖的分块 Navarro Alperin 权猜想。
Lusztig conjectured that the almost characters of a finite reductive group are up to a scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group. We propose in this paper to verify this conjecture for the uniform almost characters of the Suzuki and the Ree groups and to show that a conjecture of the same type holds for the disconnected case of type B_2. We then state a conjecture on the possible values of the uniform almost characters on unipotent elements in the disconnected case of type F_4.
Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups.
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