The intersection map of subgroups and the classes {\mathcal {PST}_c}, {\mathcal {PT}_c} and {\mathcal {T}_c}

A group G is called a ${\mathcal {T}_{c}}$ -group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes ${\mathcal {PT}_{c}}$ and ${\mathcal {PST}_{c}}$ are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that ${X=Y \cap H}$ . We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of ${\mathcal {T}_c}$ , ${\mathcal {PT}_c}$ and ${\mathcal {PST}_c}$ groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

The Dual Group of a Dense Subgroup

Throughout this abstract, G is a topological Abelian group and $\hat G$ is the space of continuous homomorphisms from G into the circle group ${\mathbb{T}}$ in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism $\hat G \to \hat D$ given by $h \mapsto h\left| D \right.$ is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D _i determines G _i with G _i compact, then $ \oplus _i D_i $ determines Π_i G _i. In particular, if each G _i is compact then $ \oplus _i G_i $ determines Π_i G _i. 3. Let G be a locally bounded group and let G ⁺ denote G with its Bohr topology. Then G is determined if and only if G ⁺ is determined. 4. Let non $\left( {\mathcal{N}} \right)$ be the least cardinal κ such that some $X \subseteq {\mathbb{T}}$ of cardinality κ has positive outer measure. No compact G with $w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$ is determined; thus if $\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$      

Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

Finite groups with some NR-subgroups or {\mathcal{H}}-subgroups

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever ${K \trianglelefteq H}$ , then ${K^G \cap H = K}$ , where K ^G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of ${\mathcal{H}}$ -subgroups: a subgroup H of G is said to be an ${\mathcal{H}}$ -subgroup of G if ${H^g \cap N_G(H) \leq H}$ for all ${g \in G}$ . In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or ${\mathcal{H}}$ -subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.     

Additive Coloring of Planar Graphs

An additive coloring of a graph G is an assignment of positive integers \({\{1,2,\ldots ,k\}}\) to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by \({\eta (G)}\) . We prove that \({\eta (G) \, \leqslant \, 468}\) for every planar graph G. This improves a previous bound \({\eta (G) \, \leqslant \, 5544}\) due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that \({\eta (G) \, \leqslant \, 36}\) for 3-colorable planar graphs, and \({\eta (G) \, \leqslant \, 4}\) for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each \({r \, \geqslant \, 2}\) there is an r-chromatic graph G _r with no additive coloring by elements of any abelian group of order r.     

Flat bundles on G/Γ and stability

Let G be a connected complex Lie group and Γ a cocompact lattice in G. Let H be a connected reductive complex affine algebraic group and \({\rho\, : \Gamma\, \longrightarrow H}\) a homomorphism such that \({\rho(\Gamma)}\) is not contained in some proper parabolic subgroup of H. Let \({E^\rho_H}\) be the holomorphic principal H–bundle on G/Γ associated to ρ. We prove that \({E^\rho_H}\) is polystable. If ρ satisfies the further condition that \({\rho(\Gamma)}\) is contained in a compact subgroup of H, then we prove that \({E^\rho_H}\) is stable.     

The reconstruction problem for certain infinite graphs

We are concerned with the notion of the degree-type (D _G ⁱ )_i∈ω of a graphG, whereD _G ⁱ is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

1. IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD _G ⁱ andD _G ⁱ⁺¹ are both finite and different from 0, thenG is reconstructible.
2. Locally finite, countably infinite graphsG, for which infinitely manyD _G ⁱ are different from 0 but only finitely manyD _G ⁱ are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.     

The Hopfian property of n-periodic products of groups

LetH be a subgroup of a groupG. A normal subgroupN _H ofH is said to be inheritably normal if there is a normal subgroup N _G of G such that N _H = N _G ∩ H. It is proved in the paper that a subgroup $N_{G_i }$ of a factor G _i of the n-periodic product Π _i∈I ⁿ G _i with nontrivial factors G _i is an inheritably normal subgroup if and only if $N_{G_i }$ contains the subgroup G _i ⁿ . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π _i∈I ⁿ G _i contains the subgroup G ⁿ . It follows that almost all n-periodic products G = G _{1 *} ⁿ G ₂ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.     

Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighbourhood Γ(u)∪Γ(v) is trivial. We find 19 finitely presented groups with the property that any such group G is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence {3,4} on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup G with $G_{uv}^{[1]} = 1$ .     

Groups whose primary subgroups are normal sensitive

A subgroup \(H\) of a group \(G\) is said to be normal sensitive in \(G\) if for every normal subgroup \(N\) of \(H, N=H\cap N^{G}\) . In this paper we study locally finite groups whose \(p\) -subgroups are normal sensitive. We show the connection between these groups and groups in which Sylow permutability is transitive.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.     

Equal Quasi-Partition of p-Groups

Let S be a subgroup of a group G. A set ${\Pi= \{H_1, \ldots , H_n\}}$ of subgroups ${H_i (i = 1, \ldots ,n)}$ with ${G=\cup_{H_i\in\Pi}H_i}$ is said to be an equal quasi-partition of G if ${H_i\cap H_j\cong S}$ and ${|H_i|=|H_j|}$ for all ${H_i, H_j\in\Pi}$ with ${i\ne j}$ . In this paper we investigate finite p-groups such that a subset of their maximal subgroups form an equal quasi-partition.     

The Chvátal–Erdös Condition for Group Connectivity in Graphs

Let G be a graph and A an abelian group with the identity element 0 and ${|A| \geq 4}$ . Let D be an orientation of G. The boundary of a function ${f: E(G) \rightarrow A}$ is the function ${\partial f: V(G) \rightarrow A}$ given by ${\partial f(v) = \sum_{e \in E^+(v)}f(e) - \sum_{e \in E^-(v)}f(e)}$ , where ${v \in V(G), E^+(v)}$ is the set of edges with tail at v and ${E^-(v)}$ is the set of edges with head at v. A graph G is A-connected if for every b: V(G) → A with ${\sum_{v \in V(G)} b(v) = 0}$ , there is a function ${f: E(G) \mapsto A-\{0\}}$ such that ${\partial f = b}$ . A graph G is A-reduced to G′ if G′ can be obtained from G by contracting A-connected subgraphs until no such subgraph left. Denote by ${\kappa^{\prime}(G)}$ and α(G) the edge connectivity and the independent number of G, respectively. In this paper, we prove that for a 2-edge-connected simple graph G, if ${\kappa^{\prime}(G) \geq \alpha(G)-1}$ , then G is A-connected or G can be A-reduced to one of the five specified graphs or G is one of the 13 specified graphs.     

Large abelian normal subgroups

In this paper we study the family of finite groups with the property that every maximal abelian normal subgroup is self-centralizing. It is well known that this family contains all finite supersolvable groups, but it also contains many other groups. In fact, every finite group G is a subgroup of some member \(\Gamma \) of this family, and we show that if G is solvable, then \(\Gamma \) can be chosen so that every abelian normal subgroup of G is contained in some self-centralizing abelian normal subgroup of \(\Gamma \).     

On embeddings of certain spherical homogeneous spaces in prime characteristic

Let $ \mathcal{G} $ be a reductive group over an algebraically closed field of characteristic p?>?0. We study embeddings of homogeneous $ \mathcal{G} $ -spaces that are induced from the G?×?G-space G, G a suitable reductive group, along a parabolic subgroup of $ \mathcal{G} $ . We give explicit formulas for the canonical divisors and for the divisors of B-semi-invariant functions. Furthermore, we show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen?CMacaulay. Examples are the G?×?G-orbits in normal reductive monoids with unit group G. Further examples are the open $ \mathcal{G} $ -orbits of the well known determinantal varieties and the varieties of (circular) complexes. Finally, we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.     

Finite-state self-similar actions of nilpotent groups

Let G be a finitely generated torsion-free nilpotent group and ${\phi:H\rightarrow G}$ be a surjective homomorphism from a subgroup H < G of finite index with trivial ${\phi}$ -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with ${(G, \phi)}$ . We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for ${(G, \phi)}$ . These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $\widehat{\phi}$ of the Lie algebra of the Mal’cev completion of G.     

{k,r – k}-Factors of r-Regular Graphs

For a set \({\mathcal{S}}\) of positive integers, a spanning subgraph F of a graph G is called an \({\mathcal{S}}\) -factor of G if \({\deg_F(x) \in \mathcal{S}}\) for all vertices x of G, where deg _F (x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an even integer such that 2 ≤ k < r/2 or an odd integer such that r/3 ≤ k < r/2. Then every r-regular graph G has a {k, r–k}-factor. Moreover, for every edge e of G, G has a {k, r–k}-factor containing e and another {k, r–k}-factor avoiding e.