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.     

Hyper-(rank one) groups

A group $G$ is called a $\mathcal{P }_1$ -group if it has a normal series of finite length whose factors have rank $1$ , while $G$ is an $\mathcal{H }_1$ -group if it has an ascending normal series of the same type. This paper investigates properties of $\mathcal{P }_1$ -groups and $\mathcal{H }_1$ -groups which correspond to known properties of nilpotent and supersoluble groups.     

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 $G$ .     

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.     

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 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.     

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.     

Geometric Satake, Springer correspondence and small representations

For a simply-connected simple algebraic group $G$ over $\mathbb C $ , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$ , generalizing a well-known fact about $GL_n$ . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.     

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.     

Derived length of solvable groups of local diffeomorphisms

Let $G$ be a solvable subgroup of the group ${\mathrm{Diff}{{\,}_{}(\mathbb{C }^{n},0)}}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$ . Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups.     

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 $p$ .     

Complemented subgroups and the structure of finite groups

Let $p$ be the smallest prime divisor of the order of a finite group $G$ . We examine the structure of $G$ under the hypothesis that $p$ -subgroups of $G$ of certain orders are complemented in $G$ . In particular, we extend some recent results.     

On the signed star domination number of regular multigraphs

Let $G$ be a graph with the vertex set $V(G)$ and the edge set $E(G)$ . A function $f: E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star dominating function of $G$ if $\sum _{e \in E_G(v)}f (e)\ge 1 $ , for every $v \in V(G)$ , where $E_G(v) = \{uv\in E(G)\,|\,u \in V (G)\}$ . The minimum values of $\sum _{e \in E_G(v)}f (e)$ , taken over all signed star dominating functions $f$ on $G$ , is called the signed star domination number of $G$ and denoted by $\gamma _{SS}(G)$ . In this paper we determine the signed star domination number of regular multigraphs.     

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.     

Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group $G$ in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of $G$ . Given a locally definable connected group $G$ (not necessarily definably generated), we prove that the $n$ -torsion subgroup of $G$ is finite and that every zero-dimensional compatible subgroup of $G$ has finite rank. Under a convexity hypothesis, we show that every zero-dimensional compatible subgroup of $G$ is finitely generated.     

Characterization of projective linear groups by the order of the normalizer of a Sylow subgroup

Let $r$ be a prime and $G$ be a finite group, and let $R, \,S$ be Sylow $r$ -subgroups of $G$ and $\text{ PGL }(2, r)$ respectively. We prove the following results: (1) If $|G|=|\text{ PGL }(2, r)|$ and $|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)} (S)|$ and $r$ is not a Mersenne prime, then $G$ is isomorphic to $\text{ PSL } (2, r) \times C_{2}, \,\text{ SL }(2, r)$ or $\text{ PGL }(2, r)$ . (2) If $|G|=|\text{ PGL }(2, r)|, \,|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)}(S)|$ where $r>3$ is a Mersenne prime and $r$ is an isolated vertex of the prime graph of $G$ , then $G\cong \text{ PGL }(2, r)$ .     

A characterization of compact complex tori via automorphism groups

We show that a compact Kähler manifold $X$ is a complex torus if both the continuous part and discrete part of some automorphism group $G$ of $X$ are infinite groups, unless $X$ is bimeromorphic to a non-trivial $G$ -equivariant fibration. Some applications to dynamics are given.     

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.     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).