Francisk Skorina Gomel State University

A subgroup R of a finite group G is called weakly subnormal in G if R is not subnormal in G but it is subnormal in every proper overgroup of R in G. In this paper, weak subnormality is used to construct a subgroup lattice of a finite soluble group containing the lattice of all subnormal subgroups. A new characterisation of Schmidt groups is also obtained: they are exactly those groups with all subgroups subnormal or weakly subnormal.

Let F and G be simple finite undirected graphs. A graph G is said to be F -irregular if any two of its distinct vertices belong to different numbers of copies of F in G . According to the strong conjecture about F -irregular graphs (Dovzhenok, Filuta, Chuhai), for any connected graph F of order at least three, there exist infinitely many F -irregular graphs. In this paper, the strong conjecture about F -irregular graphs is confirmed in the class of graphs {F} of diameter 2 . It is proved that, for every graph F of diameter 2 , there exists an infinite series of F -irregular graphs of diameter 3 .

Advanced air filtration systems have attracted significant attention due to their high efficiency, broad application scope and long-term reliability in removing air pollutants. Density functional theory (DFT) has played a crucial role in exploring the removal mechanisms for air pollutants at the atomic level, thereby enabling the rational optimization of material properties to enhance air filters performance. Herein, a multifunctional air filter (GF/CS/ZIF-8) featuring a three-dimensional interconnected network was rationally fabricated by integrating glass fiber (GF), chitosan (CS), and zeolitic imidazolate framework-8 (ZIF-8), guided by DFT simulations. The resulting composite exhibits outstanding particulate matter (PM) removal performance, achieving an efficiency of 99.56 ± 0.41 % for PM0.3, along with a high-quality factor of 0.037 ± 0.0011 Pa-1 under optimized composition (50 % GF content and 7.38 wt% ZIF-8 loading). The strong interfacial binding between CS and ZIF-8, as quantitatively confirmed by DFT calculations, underpins remarkable stability. The filter maintained high performance (98.42 ± 0.026 % for PM0.3 removal efficiency and a pressure drop of 156.7 ± 1.24 Pa) during 12 h continuous filtration, with no observable shedding of ZIF-8. In addition, the composite demonstrates exceptional volatile organic compounds (VOCs) adsorption capability, with removal efficiencies exceeding 94 %. DFT simulations further verified the feasibility of VOCs desorption, confirming the regeneration potential of the filter. Importantly, by exploiting the pH-responsive behavior of ZIF-8, the composite exhibits selective and robust antibacterial activity while maintaining excellent biosafety. Overall, the as-prepared air filter combines high PM removal efficiency, excellent VOCs adsorption capacity and superior antibacterial activity, providing a promising strategy for efficient air purification systems.

Let P_n denote a path on n vertices. A simple finite graph G is called P_n-irregular if any two distinct vertices of G belong to a different number of subgraphs of G isomorphic to P_n. Alternatively, for a fixed vertex r of P_n (the root), G is called (P_n)_r-irregular if any two distinct vertices of G act as the root r in a different number of subgraphs of G isomorphic to P_n. This paper proves that for each integer k ≥ 4, there exists an infinite family of graphs that are simultaneously P_n-irregular and (P_n)_r-irregular for every integer n satisfying 4 ≤ n ≤ k and every root r of P_n. For the path P_3, we observe that no nontrivial (P_3)_r-irregular graphs exist if r is the central vertex. In contrast, if r is an end-vertex of P_3, an infinite collection of graphs is constructed that are both P_3-irregular and (P_3)_r-irregular. In particular, these results confirm the Strong Conjecture about F-irregular graphs for the case where F is a path P_n.

Let G denote a finite group; σ={σi∣i∈I⊆{0}∪N} be some partition of the set of all primes P, where 0∈I; I be a class of finite σ0-groups that is closed under extensions, epimorphic images, and subgroups and contains all finite soluble σ0-groups. We say that a subgroup A of G is σI-subnormal in G if there is a subgroup chain A=A0≤A1≤⋯≤At=G such that either Ai−1⊴Ai, Ai/(Ai−1)Ai∈I, or Ai/(Ai−1)Ai is a σj-group for some j≠0. In this paper, we prove that the set LσI(G) of all σI-subnormal subgroups of G and some proper subsets of LσI(G) form sublattices of the lattice of all subgroups of G.