Robert M. Guralnick, Gabriel Navarro, Pham Huu Tiep
We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.
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In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.
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。
Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $π(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow graph $Γ_A(G)$ of the group $G$, with set of vertices $π(G)$, as follows: Two vertices $p,q\inπ(G)$ form an edge of $Γ_A(G)$ if either $q\inπ(A_p(G))$ or $p\in π(A_q(G))$. The following result is obtained: Theorem: Let $G$ be a finite almost simple group. Then the graph $Γ_A(G)$ is connected and has diameter at most 5. We also show how this result can be appl…
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.
On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into a…
Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay conjecture hold for such group.
The novel notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler's partition theorem.
Class-preserving Coleman Automorphisms of Finite Groups with Semidihedral Sylow 2-Subgroups
本文研究了具有半二面体Sylow 2-子群的有限群的类保持Coleman自同构群，证明其具有奇数阶，并扩展了相关文献中的结论。
We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in I}\times\mathfrak{F}_{π_i}$ for which the intersection of normalizers of all $\mathfrak{F}_i$-maximal subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$. Also we described all formations $\mathfrak{F}$ for which the intersection of $\mathfrak{F}$-subnormalizers of all Sylow (cyclic primary) subgroups of $G$ is the $\mathfrak{F}$-hypercenter …
For a group $G$ and a subgroup $H$ of $G$ this article discusses the normalizer of $H$ in the units of a group ring $RG$. We prove that $H$ is only normalized by the `obvious' units, namely products of elements of $G$ normalizing $H$ and units of $RG$ centralizing $H$, provided $H$ is cyclic. Moreover we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups and all finite groups with all Sylow subgroups being cyclic.
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。
Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal
本文分类了在互素自同构群作用下、含某不变 Sylow 子群正规化子的极大不变子群均为幂零的有限群，并证明该条件等价于所有非幂零极大不变子群均正规。
Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups
该研究证明了具有缠绕Sylow 2-子群的有限群的外自同构交集具有奇数阶，并解决了三种特定2-群的正规化问题。
In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. Unrefinable partitions into distinct parts are those in which no part x can be replaced with integers whose sum is x obtaining a new partition into distinct parts. We prove here that the (n-1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition …
On the number of $\ell$-regular conjugacy classes
本文证明了当素数 $\ell>5$ 时，有限群的 $\ell$-正则共轭类数不少于其 Sylow $\ell$-子群正规化子的相应数量。
On the local representation theory of symmetric groups
本文研究对称群Sylow p-子群的正规化子对其不可约特征标的作用，通过建立特征标与显式函数等价类的一一对应来描述该作用，并适用于伽罗瓦作用。
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.
We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.
On an extension of Shlyk's theorem
研究将 Shlyk 定理推广至含某 Sylow 子群正规化子的非可解群，证明其所有非幂零极大子群的交具有幂零性。
We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups.
We use binary trees to study the Bratteli diagram of Sylow 2-subgroups of symmetric groups. We show that it is simple, has a recursive structure, and self-similarities at all scales. We contrast its subgraph of one-dimensional representations with the Macdonald tree. We exploit the recursive structure to find the multiplicities of irreducible characters in the restriction to a Sylow 2-subgroup of odd-dimensional representations of the symmetric group $S_{2^k}$.
Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the …
Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup $P_n$. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra $\mathcal{H}(\mathfrak{S}_n, P_n, 1_{P_n})$.
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 $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^…
We examine $p$-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a $p$-group $G$, when $p=2$, the center of $G$ has index at most 16 and when $p$ is odd the center of $G$ has index at most $p^3$.
On the density of Sylow numbers
本文研究了Sylow p-数的密度，证明当p为奇数时，Sylow p-数的自然密度为0。
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.
We construct a natural bijection between odd-degree irreducible characters of S_n and linear characters of its Sylow 2-subgroup P_n. When n is a power of 2, we show that such a bijection is nicely induced by the restriction functor. We conclude with a characterization of the irreducible characters χof S_n such that the restriction of χto P_n has a unique linear constituent.
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 $G$.
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…
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.
The Euler characteristic of an endotrivial complex
本文研究了有限群G在特征p的域k下的端 trivial 复合体群的欧拉特征，探讨了Lefschetz同态的性质及其满射条件，以及核的确定。
For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reduced the McKay--Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay--Navarro (also c…
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.
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 McKay Conjecture on character degrees
本文证明了关于特征标阶数的McKay猜想，即任意有限群中阶与素数互素的不可约复特征标个数等于其Sylow子群正规化子中的个数，通过完成Isaacs--Malle--Navarro对有限单群表示猜想的归约证明，并引入F-稳定约化子群作为有效替代来研究其特征标。
Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of both of these conjectures are true.
Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do so, we, on one hand, make use of the recent proof of the McKay conjecture for the prime 2 by Malle and Späth, and, on the other hand, prove lower bounds for the class number of the semidirect product of an odd-order group acting on an abelian $2$-group.
We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.
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。
A group $G$ is said to be cut if, for every $g \in G$, each generator of $< \! g \! >$ is conjugated to either $g$ or $g^{-1}$. It is conjectured that a Sylow 3-subgroup $P$ of a cut group $G$ is cut. We prove that this is true if $|G|$ is odd.
Exotic Fusion Systems Related to Sporadic Simple Groups (Version With Code)
本文借助 MAGMA 计算，对与散在单群 Co_1、F_3 和 M 在奇素数处相关的奇异及饱和融合系统进行分类。
The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those groups of order 64 with an odd-order automorphism. In view of the fact that we already had determined these groups and that these automorphism groups are not given explicitly in the literature, we have appended to this report these automorphism groups. In another project we were looking for new complete groups by following automorphism group towers up to comple…
Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.
On the number and sizes of double cosets of Sylow subgroups of the symmetric group
研究对称群 Sylow $p$-子群双陪集的数量与大小，证明 $p$ 为奇数且 $n$ 较大时大多数双陪集达最大规模，并发现除少数小例外所有可能大小均出现。
We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator …
On the degrees of irreducible characters fixed by some field automorphism, p-solvable groups
本文将实值不可约特征标奇次数蕴含正规Sylow 2-子群的结论推广到p-可解群，证明若p不整除被p阶域自同构固定的不可约特征标次数，则该群具有正规Sylow p-子群。
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.
Sylow synchronization in finite groups: the good case
本文证明有限可解群在特定Sylow子群交集条件下存在同步化元素，覆盖奇数阶群并统一旧结果，同时完成对称群、交错群及单群猜想的证明。
Hopf Galois structures, skew braces for groups of size $p^nq$: The cyclic Sylow subgroup case
本文研究了阶为$p^nq$且Sylow-$p$-子群为循环群的群上的Hopf-Galois结构，枚举了Galois $G$-扩张上类型为$N$的结构，计算了对应的斜brace数量，并在$q<p$时给出了完整分类。
In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.
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。
Coprime automorphisms of profinite groups and the commuting probability of Sylow subgroups
研究 profinite 群中 Sylow 子群交换概率为正的条件，证明在互素自同构作用下，若特定 Sylow 子群交换概率为正，则群是 virtually pro-p-可解的，并给出非 p-可解长度的界。
Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the L-theory of virtually cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2…
New classes of IYB groups
该研究通过证明两类满足西罗子群幂零类约束的有限群为IYB群，推广了已有结论，还得到两类有限群同构于幂零型斜左brace乘法群的结果，推进了IYB群的公开问题研究。
In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these methods to characterize $\alt(9)$. Furthermore, we describe a particular hypothesis related to the 3-local structure of finite groups of local characteristic 3 and characterize two almost simple proper extensions of $Ω^+_8(2)$ as examples of the local approach to group recognition. We also give a 3-local characterization of the sporadic simple group $\HN$ us…
Skew braces of size $p^2q^2$. The cyclic Sylow $q$-subgroup case
本文证明互素阶群的正规子群性质可推广到斜括号，给出分类方法，并应用于确定 $p^2q^2$ 阶非阿贝尔型斜括号。
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}.
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