A subgroup H of a finite group G is said to be conjugate-permutable if HHg = HgH for all g\in G. More generaly, if we limit the element g to a subgroup R of G, then we say that the subgroup H is R-conjugate-permutable. By means of the R-conjugatepermutable subgroups, we investigate the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B under the condition that G = AB, where A and B are subgroups of G. Some results known in the literature are improved and generalized in the paper., Xianhe Zhao, Ruifang Chen., and Obsahuje seznam literatury
We introduce the rainbowness of a polyhedron as the minimum number $k$ such that any colouring of vertices of the polyhedron using at least $k$ colours involves a face all vertices of which have different colours. We determine the rainbowness of Platonic solids, prisms, antiprisms and ten Archimedean solids. For the remaining three Archimedean solids this parameter is estimated.
Let T, T′ be weak contractions (in the sense of Sz.-Nagy and Foia¸s), m, m′ the minimal functions of their C0 parts and let d be the greatest common inner divisor of m, m′ . It is proved that the space I(T, T′ ) of all operators intertwining T, T′ is reflexive if and only if the model operator S(d) is reflexive. Here S(d) means the compression of the unilateral shift onto the space H 2 ⊖dH2 . In particular, in finite-dimensional spaces the space I(T, T′ ) is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of T, T′ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of I(T, T′ ).
We extend a result of Rangaswamy about regularity of endomorphism rings of Abelian groups to arbitrary topological Abelian groups. Regularity of discrete quasi-injective modules over compact rings modulo radical is proved. A characterization of torsion LCA groups $A$ for which ${\rm End}_c(A)$ is regular is given.
A subset Y of a space X is almost countably compact in X if for every countable cover U of Y by open subsets of X, there exists a finite subfamily V of U such that Y ⊆ U V . In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.
A subspace Y of a space X is almost Lindelöf (strongly almost Lindelöf) in X if for every open cover U of X (of Y by open subsets V of X), there exists a countable subset V of U such that Y ⊆ S {V : V ∈ V }. In this paper we investigate the relationships between relatively almost Lindelöf subset and relatively strongly almost Lindelöf subset by giving some examples, and also study various properties of relatively almost Lindelöf subsets and relatively strongly almost Lindelöf subsets.
Simple modules for restricted Lie superalgebras are studied. The indecomposability of baby Kac modules and baby Verma modules is proved in some situation. In particular, for the classical Lie superalgebra of type $A(n|0)$, the baby Verma modules $Z_{\chi }(\lambda )$ are proved to be simple for any regular nilpotent $p$-character $\chi $ and typical weight $\lambda $. Moreover, we obtain the dimension formulas for projective covers of simple modules with $p$-characters of standard Levi form.