We give a condition on a full coaction (A,G, δ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A×δ| G/N is Morita equivalent to A×δ G×δ̂,r N . This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raebur...

We consider decomposition spaces R/G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R/G constructed via modular embeddings of R/G into a Euclidean space promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R/G×R by R for any m ≥ 0 imposes quantitative topological con...

Let G be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of G/H to be embeddable into an irreducible G-module. If H is reductive we also find a necessary and sufficient condition for a closed embedding of G/H into an irreducible module to exist. These conditions are stated in terms of the group of central automorphisms of G/H .

A cycle C in a graph G is said to be dominating if E(G−C) = ∅. Enomoto et al. showed that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 2, then every longest cycle is dominating. But it is unknown whether the condition on the independence number is sharp. In this paper, we show that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 1, then G has a longest cycle which...

For a graph G, we denote by i(G) the number of isolated vertices of G. We prove that for a connected graph G of order at least five, if i(G − S) < |S| for all ∅ = S ⊆ V (G), then G has a spanning tree T such that the distance in T between any two leaves of T is at least four. This result was conjectured by Kaneko in ”Spanning trees with constrains on the leaf degree”, Discrete Applied Math., 11...

We study an RG-module A , where R is a ring, A/CA(G) is infinite, CG(A) = 1, G is a group. Let Lnf(G) be the system of all subgroups H ≤ G such that the quotient modules A/CA(H) are infinite. We investigate an RG-module A such that Lnf(G) satisfies either the weak minimal condition or the weak maximal condition as an ordered set. It is proved that if G is a locally finite group then either G is...

A subgroup A of the ^-primary group G is nice if p(G/A) = {pG, A }/A for all ordinals a. We consider the following countability condition: there exists a collection G of nice subgroups of G such that (0) oee. (1) 6 is closed with respect to group-theoretic union in G. (2) If A^e and if H is a subgroup of G such that {Af H}/A is countable, there exists i ? £ G such that 5 2 {-4, H\ and such that...

‎Let $tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$‎. ‎Let $bar{G}=G/H_{G}$ and $bar{H}=H/H_{G}$‎. ‎We say that $H$ is $Phi$-$tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $bar{T}$ of $bar{G}$ and some $tau$-subgroup $bar{S}$ of $bar{G}$ contained in $bar{H}$‎, ‎$bar{H}bar{T}$ is $S$-quasinormal in $bar{G}$ and $bar{H}capbar{T}leq bar{S}Phi(bar{H})$‎. ‎I...

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K, we give a sufficient condition for a valued subfield of the field of generalized power series F ((G)) to admit aK-valuation basis. We show that the field of rational functions F (G) and the field F (G) of power series in F ((G))...