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All groups under consideration are finite. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G be a group, F be a class of groups, σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } , and σ (F) = ∪ G ∈ F σ (G). A function f of the form f : σ → { formations of groups } is called a formation σ-function. For any formation σ-function f the class L F σ (f) is defined as follows: L F σ (f) = (G is a group |G = 1 or G ≠ 1 and G/O σ i ′ , σ i (G) ∈ f (σ i) for all σ i ∈ σ (G)). If for some formation σ-function f we have F = L F σ (f) , then the class F is called σ-local and f is called a σ-local definition of F. Every formation is called 0-multiply σ -local. For n > 0 , a formation F is called n-multiply σ-local provided either F = (1) is the class of all identity groups or F = L F σ (f) , where f (σ i) is (n − 1) -multiply σ-local for all σ i ∈ σ (F). Let τ (G) be a set of subgroups of G such that G ∈ τ (G) . Then τ is called a subgroup functor if for every epimorphism φ : A → B and any groups H ∈ τ (A) and T ∈ τ (B) we have H φ ∈ τ (B) and T φ − 1 ∈ τ (A) . A formation of groups F is called τ-closed if τ (G) ⊆ F for all G ∈ F . A complete lattice of formations θ is called separable, if for any term ν (x 1 , ... , x m) signatures { ∩ , ∨ θ } , any θ-formations F 1 , ... , F m and any group A ∈ ν (F 1 , ... , F m) there are groups A 1 ∈ F 1 , ... , A m ∈ F m such that A ∈ ν (θ form A 1 , ... , θ form A m) . We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations. [ABSTRACT FROM AUTHOR]

Let 휎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 휎-subnormal subgroup is modular. [ABSTRACT FROM AUTHOR]

The Sichuan-Tibet traffic corridor (STTC) is a critical transportation route in China that spans the western Sichuan Basin and the Tibetan Plateau. This area has a complex geological history and presents challenges for infrastructure construction due to its high seismic intensity and extreme elevation. Recent research has made advancements in addressing these challenges and supporting infrastructure development and mineral resource exploitation. The second volume of this Research Topic focuses on active faults, magmatic-hydrothermal activities, and the tectonic evolution history of the Tibetan Plateau within the STTC. Additionally, it explores the diverse range of mineral resources in the area. The findings from these studies will contribute to further geological investigations and resource assessments in the STTC. [Extracted from the article]

Throughout this paper, all groups are finite and G always denotes a group. Let σ = { σ i | i ∈ I } be some partition of the set of all primes ℙ , be a class of groups, σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } , and σ () = ⋃ G ∈ σ (G). A function f of the form f : σ → { formations of groups } is called a formation σ -function. For any formation σ -function f the class L F σ (f) is defined as follows: L F σ (f) = (G | G = 1  or  G ≠ 1  and  G / O σ i ′ , σ i (G) ∈ f (σ i)  for all  σ i ∈ σ (G)). If for some formation σ -function f we have = L F σ (f) , then the class is called σ -local and f is called a σ -local definition of . If, in addition, f (σ i) ⊆ L F σ (f) for all i , then f is called integrated. Every formation is called 0-multiply σ -local. For n > 0 , a formation is called n -multiply σ -local provided either = (1) is the class of all identity groups or = L F σ (f) , where f (σ i) is (n − 1) -multiply σ -local for all σ i ∈ σ (). Let τ (G) be a set of subgroups of G such that G ∈ τ (G). Then τ is called a subgroup functor if for every epimorphism φ : A → B and any groups H ∈ τ (A) and T ∈ τ (B) we have H φ ∈ τ (B) and T φ − 1 ∈ τ (A). A class of groups is called τ -closed if τ (G) ⊆ for all G ∈ . Let be a complete lattice of formation. Then the symbol σ denotes the set of all formations such that = L F σ (f) , where f (σ i) ∈ for all σ i ∈ σ (). A complete lattice σ we call σ -inductive, if for any set { j | j ∈ J } ⊆ σ and for any set { f j | j ∈ J } , where j = L F σ (f j) , f j is an integrated, and f j (σ i) ∈ for all σ i ∈ σ () , we have σ form  (⋃ j ∈ J j) = L F σ (f) , where σ form  (⋃ j ∈ J j) is the intersection of all formations from σ containing ⋃ j ∈ J j and f is a formation σ -function such that for all σ i ∈ σ (⋃ j ∈ J j) f (σ i) = form  (⋃ j ∈ J f j (σ i)) is the intersection of all formations from containing ⋃ j ∈ J f j (σ i). We prove that the set of all τ -closed n -multiply σ -local formations forms a complete σ -inductive lattice of formations. [ABSTRACT FROM AUTHOR]

Let X be a class of groups. Suppose that with each group G ∈ X we associate some system of its subgroups τ (G) . Then τ is said to be a subgroup functor (in the sense of Skiba) on X if the following conditions are hold: (1) G ∈ τ (G) for each group G ∈ X ; (2) for any epimorphism φ : A → B , where A , B ∈ X , and for any groups H ∈ τ (A) and T ∈ τ (B) we have H φ ∈ τ (B) and T φ − 1 ∈ τ (A) . In this paper, we discuss some applications of such subgroup functors in the theory of generalized T-groups. [ABSTRACT FROM AUTHOR]

In this paper, G is a finite group and σ a partition of the set of all primes ℙ, that is, σ = {σi| i ∈ I}, where ℙ = ⋃i∈I σi and σi ∩ σj = ∅ for all i = j; σ (G) = {σi| σi ∩ π(G) = ∅}. We say that G is a σ-tower group if either G = 1 or G has a normal series 1 = G0< G1 < · · · < Gt−1 < Gt= G such that Gk/Gk−1 is a σi-group, σi ∈ σ(G), and G/Gk and Gk−1 are σ′i-groups for all k = 1,... , t. Now we associate with each group G = 1 a directed graph Γ = ΓH σ (G) as follows: Let σ(G) = {σ1 , σ2 ,... , σt}. ΓH σ (G) has t vertices σ1 , σ2 ,... , σt and (σi , σj) is an arc of Γ (directed from σi to σj) if and only if. And we call such a graph the Hawkes σ-graph of G. In this paper, we study various properties of finite Sylow tower and σ-tower groups. In particular, we prove that G is a σ-tower group if and only if the Hawkes σ-graph of G has no circuits, and we describe the structure of a σ-tower group G by appealing to some other properties of the graph ΓHσ (G). [ABSTRACT FROM AUTHOR]

This paper reviews published and presents new data on U-Pb detrital zircon ages, and petrographic, geochemical and isotope (Sm-Nd, Lu-Hf) compositions obtained from greywacke sandstones of Kazakhstan in order to reconstruct fossil intra-oceanic arcs that once existed at Pacific-type convergent margins of the Paleo-Asian Ocean (PAO) in Paleozoic time. We focus on orogenic belts of central Kazakhstan (Itmurundy and Tekturmas) and eastern Kazakhstan (Zharma and Char) in the western Central Asian Orogenic belt. These orogenic belts host accretionary complexes with greywacke sandstones of early Paleozoic (central Kazakhstan) and middle-late Paleozoic (eastern Kazakhstan) ages. First, we evaluate general perspectives for studying sandstones to reconstruct survived and disappeared magmatic arcs, taking into account episodes of subduction erosion. Then we discuss the analytical data from sandstones to make conclusions about the ages and formation settings of their igneous protoliths and define maximum deposition ages. Finally, we discuss the role of serpentinite mélanges in tectonic reconstructions. We argue that sandstones hosted by accretionary complexes are typically greywackes deposited close to their igneous sources and buried rapidly. The provenances of the studied greywacke sandstones of central and eastern Kazakhstan were dominated by mafic to andesitic igneous protoliths derived from juvenile mantle sources. The igneous rocks in the provenances were emplaced in an intra-oceanic arc setting. The sandstones were deposited in fore-arc/trench basins or, to a lesser degree, in back-arc basins. The data from both sandstones and serpentinite mélanges reconstruct middle-late-Cambrian, Ordovician, late-Devonian and Carboniferous arcs of the western PAO. The middle-late Cambrian arcs were fully destroyed by subduction erosion, whereas the Ordovician and Carboniferous arcs survived. The late-Devonian arcs were also eroded, but partly. Both the early and late Paleozoic active margins of the PAO were characterized by alternating periods of accretionary growth and subduction erosion. [ABSTRACT FROM AUTHOR]