Brauer characters and rationality

A finite group $G$ has no non-trivial rational-valued irreducible $p$ -Brauer characters if and only if $G$ has no non-trivial rational elements of order not divisible by $p$ .     

Minimal subgroups and the structure of finite groups

A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$ . In this paper we prove that a finite group $G$ is $p$ -nilpotent if every minimal subgroup of $P\bigcap G^{N}$ is weakly-supplemented in $G$ , and when $p=2$ either every cyclic subgroup of $P\bigcap G^{N}$ with order 4 is weakly-supplemented in $G$ or $P$ is quaternion-free, where $p$ is the smallest prime number dividing the order of $G$ , $P$ a sylow $p$ -subgroup of $G$ .     

Structure of p-complements in groups with three p-regular class sizes

Let $G$ be a finite $p$ -solvable group for some prime $p$ and suppose that the set of $p$ -regular conjugacy class sizes is $\{1, m, mn\}$ with $(m, n)=1$ and $m$ coprime to $p$ . We show that $m=q^b$ for some prime $q$ and we describe the structure of the $p$ -complements of $G$ .     

Subgroup properties of pro-p extensions of centralizers

We prove that a finitely generated pro- $p$ group acting on a pro- $p$ tree $T$ with procyclic edge stabilizers is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with vertex groups being stabilizers of certain vertices of $T$ and edge groups (when non-trivial) being stabilizers of certain edges of $T$ , in the following two situations: (1) the action is $n$ -acylindrical, i.e., any non-identity element fixes not more than $n$ edges; (2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro- $p$ groups from the class $\mathcal L $ defined and studied in Kochloukova and Zalesskii (Math Z 267:109–128, 2011) as pro- $p$ analogues of limit groups. We prove that every pro- $p$ group $G$ from the class $\mathcal L $ is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class $\mathcal L $ of lower level than $G$ with respect to the natural hierarchy. This allows us to give an affirmative answer to questions 9.1 and 9.3 in Kochloukova and Zalesskii (Math Z 267:109–128, 2011). Namely, we prove that a group $G$ from the class $\mathcal L $ has Euler–Poincaré characteristic zero if and only if it is abelian, and if every abelian pro- $p$ subgroup of $G$ is procyclic and $G$ itself is not procyclic, then $\mathrm{def}(G)\ge 2$ . Moreover, we prove that $G$ satisfies the Greenberg–Stallings property and any finitely generated non-abelian subgroup of $G$ has finite index in its commensurator.     

On S-permutably embedded subgroups of finite groups

A subgroup $A$ of a finite group $G$ is said to be $S$ -permutably embedded in $G$ if for each prime $p$ dividing the order of $A$ , every Sylow $p$ -subgroup of $A$ is a Sylow $p$ -subgroup of some $S$ -permutable subgroup of $G$ . In this paper we determine how the $S$ -permutable embedding of several families of subgroups of a finite group influences its structure.     

p-Groups with few conjugacy classes of normalizers

For a group $G$ , denote by $\omega (G)$ the number of conjugacy classes of normalizers of subgroups of $G$ . Clearly, $\omega (G)=1$ if and only if $G$ is a Dedekind group. Hence if $G$ is a 2-group, then $G$ is nilpotent of class $\le 2$ and if $G$ is a $p$ -group, $p>2$ , then $G$ is abelian. We prove a generalization of this. Let $G$ be a finite $p$ -group with $\omega (G)\le p+1$ . If $p=2$ , then $G$ is of class $\le 3$ ; if $p>2$ , then $G$ is of class $\le 2$ .     

On cyclic fixed points of spectra

For a finite $p$ -group $G$ and a bounded below $G$ -spectrum $X$ of finite type mod  $p$ , the $G$ -equivariant Segal conjecture for $X$ asserts that the canonical map $X^G \rightarrow X^{hG}$ , from $G$ -fixed points to $G$ -homotopy fixed points, is a $p$ -adic equivalence. Let $C_{p^n}$ be the cyclic group of order  $p^n$ . We show that if the $C_p$ -equivariant Segal conjecture holds for a $C_{p^n}$ -spectrum $X$ , as well as for each of its geometric fixed point spectra $\varPhi ^{C_{p^e}}(X)$ for $0 < e < n$ , then the $C_{p^n}$ -equivariant Segal conjecture holds for  $X$ . Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.     

On some primary subgroups and the structure of finite groups

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$ . $H$ is said to be an $s$ -quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$ , a Sylow $p$ -subgroup of $H$ is also a Sylow $p$ -subgroup of some $S$ -quasinormal subgroup of $G$ ; $H$ is said to be $c$ -normal in $G$ if $G$ has a normal subgroup $T$ such that $G=HT$ and $H\cap T\le H_{G}$ , where $H_{G}$ is the normal core of $H$ in $G$ . We fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1<|D|<|P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$ -quasinormally embedded or $c$ -normal in $G$ . Some recent results are generalized and unified.     

SK_1 and Lie algebras

We investigate the vanishing of the group $SK_1(\Lambda (G))$ for the Iwasawa algebra $\Lambda (G)$ of a pro- $p$ $p$ -adic Lie group $G$ (with $p \ne 2$ ). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.     

Finite groups all of whose second maximal subgroups are {\mathcal{H }}_p-groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is called an $\mathcal{H }$ -subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$ . A group $G$ is said to be an ${\mathcal{H }}_p$ -group if every cyclic subgroup of $G$ of prime order or order 4 is an $\mathcal{H }$ -subgroup of $G$ . In this paper, the structure of a finite group all of whose second maximal subgroups are ${\mathcal{H }}_p$ -subgroups has been characterized.     

On minimal non-M_rC-groups

A group $G$ is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let $D(G)$ denotes the subgroup of $G$ generated by all the Chernikov divisible normal subgroups of $G$ . If $G$ is a soluble-by-finite minimax group and if $D(G)=1$ , then $G$ is said to be a reduced minimax group. Also $G$ is said to be an $ M_{r}C$ -group (respectively, $PC$ -group), if $G/C_{G} \left(x^{G}\right)$ is a reduced minimax (respectively, polycyclic-by-finite) group for all $x\in G$ . These are generalisations of the familiar property of being an $FC$ -group. Finally, if $\mathfrak X $ is a class of groups, then $G$ is said to be a minimal non- $\mathfrak X $ -group if it is not an $\mathfrak X $ -group but all of whose proper subgroups are $\mathfrak X $ -groups. Belyaev and Sesekin characterized minimal non- $FC$ -groups when they have a non-trivial finite or abelian factor group. Here we prove that if $G$ is a group that has a proper subgroup of finite index, then $G$ is a minimal non- $M_{r}C$ -group (respectively, non- $PC$ -group) if, and only if, $G$ is a minimal non- $FC$ -group.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

Fourier analysis of subgroup conjugacy invariant functions on finite groups

Given a finite group $G$ and a subgroup $H\le G$ , we develop a Fourier analysis for $H$ -conjugacy invariant functions on $G$ , without the assumption that $H$ is a multiplicity-free subgroup of $G$ . We also study the Fourier transform for functions in the center of the algebra of $H$ -conjugacy invariant functions on $G$ . We show that a recent calculation of Cesi is indeed a Fourier transform of a function in the center of the algebra of functions on the symmetric group that are conjugacy invariant with respect to a Young subgroup.     

Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions

Let $G$ be a bounded Jordan domain in the complex plane. The Bergman polynomials $\{p_n\}_{n=0}^\infty $ of $G$ are the orthonormal polynomials with respect to the area measure over $G$ . They are uniquely defined by the entries of an infinite upper Hessenberg matrix $M$ . This matrix represents the Bergman shift operator of $G$ . The main purpose of the paper is to describe and analyze a close relation between $M$ and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of $\overline{G}$ . Our results are based on the strong asymptotics of $p_n$ . As an application, we describe and analyze an algorithm for recovering the shape of $G$ from its area moments.     

Modular categories associated with unipotent groups

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$ , the $\overline{\mathbb{Q }}_l$ -linear triangulated braided monoidal category of $G$ -equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$ -complexes on $G$ under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to $G$ and $e$ a modular category $\mathcal{M }_{G,e}$ . In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$ .     

Feebly compact paratopological groups and real-valued functions

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group $G$ can fail to be a topological group. Our group $G$ has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group $G$ all countable subsets of which are closed. Another peculiarity of the group $G$ is that it contains a nonempty open subsemigroup $C$ such that $C^{-1}$ is closed and discrete, i.e., the inversion in $G$ is extremely discontinuous. We also prove that for every continuous real-valued function $g$ on a feebly compact paratopological group $G$ , one can find a continuous homomorphism $\varphi $ of $G$ onto a second countable Hausdorff topological group $H$ and a continuous real-valued function $h$ on $H$ such that $g=h\circ \varphi $ . In particular, every feebly compact paratopological group is $\mathbb{R }_3$ -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.     

Character formulæ and GKRS multiplets in equivariant K-theory

Let $G$ be a compact Lie group, $H$ a closed subgroup of maximal rank and $X$ a topological $G$ -space. We obtain a variety of results concerning the structure of the $H$ -equivariant K-ring $K_H^*(X)$ viewed as a module over the $G$ -equivariant K-ring $K_G^*(X)$ . One result is that the module has a nonsingular bilinear pairing; another is that the module contains multiplets which are analogous to the Gross–Kostant–Ramond–Sternberg multiplets of representation theory.     

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

Chromatic Gallai identities operating on Lovász number

If $G$ is a triangle-free graph, then two Gallai identities can be written as $\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)$ , where $\alpha $ and $\overline{\chi }$ denote the stability number and the clique-partition number, and $L(G)$ is the line graph of  $G$ . We show that, surprisingly, both equalities can be preserved for any graph $G$ by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of  $G$ . As a consequence, one obtains an operator $\Phi $ which associates to any graph parameter $\beta $ such that $\alpha (G) \le \beta (G) \le \overline{\chi }(G)$ for all graph $G$ , a graph parameter $\Phi _\beta $ such that $\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)$ for all graph $G$ . We prove that $\vartheta (G) \le \Phi _\vartheta (G)$ and that $\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)$ for all graph  $G$ , where $\vartheta $ is Lovász theta function and $\overline{\chi }_f$ is the fractional clique-partition number. Moreover, $\overline{\chi }_f(G) \le \Phi _\vartheta (G)$ for triangle-free $G$ . Comparing to the previous strengthenings $\Psi _\vartheta $ and $\vartheta ^{+ \triangle }$ of $\vartheta $ , numerical experiments show that $\Phi _\vartheta $ is a significant better lower bound for $\overline{\chi }$ than $\vartheta $ .