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.     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

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 $ .     

On locally finite groups with a four-subgroup whose centralizer is small

Let $G$ be a locally finite group which contains a non-cyclic subgroup $V$ of order four such that $C_{G}\left( V\right) $ is finite and $C_{G}\left( \phi \right)$ has finite exponent for some $\phi \in V$ . We show that $[G,\phi ]^{\prime }$ has finite exponent. This enables us to deduce that $G$ has a normal series $1\le G_1\le G_2\le G_3\le G$ such that $G_1$ and $G/G_2$ have finite exponents while $G_2/G_1$ is abelian. Moreover $G_3$ is hyperabelian and has finite index in $G$ .     

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 $\le 2$ .     

Weak quadratic overgroups for a class of Lie groups

Let $G$ be a connected and simply connected Lie group with Lie algebra $\mathfrak g $ . We say that a subset $X$ in the set $\mathfrak g ^\star / G$ of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in $X$ . In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup $G^+$ and a quadratic map $\varphi $ sending each generic orbit in $\mathfrak g ^\star $ to a $G^+$ -orbit in $\mathfrak{g ^+}^\star $ , in such a manner that the set $\varphi (\mathfrak g ^\star _{gen}){/ G^+}$ is convex hull separable. We then call $G^+$ a weak quadratic overgroup for $G$ . Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.     

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.     

Normality and smoothness of simple linear group compactifications

Given a semisimple algebraic group $G$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $G\times G$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $\mathbb{P }(\mathrm{End}(V))$ , where $V$ is a finite dimensional rational $G$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $V$ . In particular, we show that ${\mathrm{Sp}}(2r)$ (with $r \geqslant 1$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$      

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.     

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 $\mathfrak{C }_p^{\pm }$ .     

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

A group $G$ is said to be periodic if for every $g\in G$ there exists a positive integer $n$ with $g^n=\mathrm{Id}$ . We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a probability measure $\mu $ is finite. Moreover if the group consists of homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T }^2$ . In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Volume invariant and maximal representations of discrete subgroups of Lie groups

Let $\Gamma $ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma $ into $G$ , which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma $ into $G$ .     

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.     

The topology of parabolic character varieties of free groups

Let $G$ be a complex affine algebraic reductive group, and let $K\,\subset \, G$ be a maximal compact subgroup. Fix h $\,:=\,(h_{1}\,,\ldots \,,h_{m})\,\in \, K^{m}$ . For $n\, \ge \, 0$ , let $\mathsf X _{\mathbf{{h}},n}^{G}$ (respectively, $\mathsf X _{\mathbf{{h}},n}^{K}$ ) be the space of equivalence classes of representations of the free group on $m+n$ generators in $G$ (respectively, $K$ ) such that for each $1\le i\le m$ , the image of the $i$ -th free generator is conjugate to $h_{i}$ . These spaces are parabolic analogues of character varieties of free groups. We prove that $\mathsf X _{\mathbf{{h}},n}^{K}$ is a strong deformation retraction of $\mathsf X _{\mathbf{{h}},n}^{G}$ . In particular, $\mathsf X _{\mathbf{{h}},n}^{G}$ and $\mathsf X _{\mathbf{{h}},n}^{K}$ are homotopy equivalent. We also describe explicit examples relating $\mathsf X _{\mathbf{{h}},n}^{G}$ to relative character varieties.     

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 $G\cong \text{ PGL }(2, r)$ .     

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$ , and let $\mu :\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$ , $O(V)$ , or $Sp(V)$ . For these classes of examples, $\mu ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $\mu ^{-1}(0)/\!/G$ .     

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 $\mathbb{C }^{k+1}$ ).     

On the Degenerate Crossing Number

The degenerate crossing number ${\text{ cr}^{*}}(G)$ of a graph $G$ is the minimum number of crossing points of edges in any drawing of $G$ as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph $G$ with $n$ vertices and $e \ge 4n$ edges ${\text{ cr}^{*}}(G)=\Omega \big (e^4 / n^4\big )$ . In this paper we completely resolve the main open question about degenerate crossing numbers and show that ${\text{ cr}^{*}}(G)=\Omega \big (e^3 / n^2 \big )$ , provided that $e \ge 4n$ . This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.