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Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G a finite group and σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } . A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i and H contains exactly one Hall σi-subgroup of G for every σ i ∈ σ (G) . Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H -permutable if HA = AH for every member A ∈ H ; m- H -permutable if H = 〈 A , B 〉 for some modular subgroup A and H -permutable subgroup B of G; weakly m- H -permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H for some m- H -permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m- H -permutable in G. Some recent results are generalized and unified. [ABSTRACT FROM AUTHOR]

A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. A subgroup A of a group G is called OS-propermutablein G if there is a subgroup B such that G = N G (A) B , where AB is a subgroup of G and A permutes with all Schmidt subgroups of B. We proved p -solubility of a group in which a Sylow p -subgroup is OS-propermutable, where p ≥ 7 7. For p < 7 all non-Abelian composition factors of such group are listed. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π -property in G if there exists a chief series Γ G : 1 = G 0 < G 1 < · · · < G n = G of G such that for every G-chief factor G i / G i − 1 (1 ≤ i ≤ n) of Γ G , | G / G i − 1 : N G / G i − 1 (H G i − 1 / G i − 1 ∩ G i / G i − 1) | is a π (H G i − 1 / G i − 1 ∩ G i / G i − 1) -number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial Π -property on the structure of a finite group. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. We say that H has an IC-property if the intersection of H and [ H , G ] satisfy some conditions. In this paper, we investigate the structure of a finite group G under the assumption that some maximal subgroups, 2-maximal subgroups, 3-maximal subgroups of G have the ICp-property and obtain some new characterizations of p-solvable and solvable groups. [ABSTRACT FROM AUTHOR]

AbstractA subgroup S of a finite group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we investigate pF -hypercentrally embedded structure of finite groups by using p-sylowizers of some p-subgroups with fixed order. [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ -subnormal in G if there is a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every i = 1 , ... , n , H i - 1 normal in H i or H i / (H i - 1) H i is a σ i -group for some i ∈ I . A subgroup H of G is said to be weakly σ -n-embedded in G, if there is a σ -subnormal subgroup T of G such that H G = H T and H ∩ T ≤ H σ G , where the subgroup H σ G is generated by all σ -subnormal subgroups of G contained in H. In this paper, we study the properties of weakly σ -n-embedded subgroups and use them to determine the structure of finite groups. Some known results are gerneralized. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be a partition of the set of all primes. A subgroup H of a finite group G is called σ -subnormal in G if there exists a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G such that for every j = 1 , 2 , ... , n either H j - 1 is normal in H j or H j / C o r e H j (H j - 1) is a σ i -subgroup for some i ∈ I . Let π be a set of primes and G ∈ E π . A subgroup H of G is said to be π -subnormal in G if H ∩ S is a Hall π -subgroup of H for any Hall π -subgroup S of G. In this paper, the class of all finite groups in which all σ i -subnormal subgroups form a join-semilattice, for any i ∈ I , is characterized. Furthermore, for such groups, the authors investigate the following Kegel–Wielandt σ -problem: Is it true that a subgroupHis σ -subnormal inGifHis σ i -subnormal inGfor all i ∈ I ? [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure HG of H in G. In this paper, we shall obtain some characterizations about p-nilpotency of G under the assumption that certain some subgroups of G of prime power order are Hall normally embedded in G. [ABSTRACT FROM AUTHOR]

Let G N be the nilpotent residual of a group G. Then L(G) is the set of all elements in G which induce power automorphisms on G N by conjugation, and G is called an L -group if G = L (G) . In this paper, L(G) and L -groups are studied. In particular, the structures of L -groups and minimal non- L -groups are described. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies the Π-property in G if for any chief factor L/K of G, ∣G/K: NG/K(HK/K∩ L/K)∣ is a π(HK/K∩L/K)-number. We obtain some criteria for the p-supersolubility or p-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the Π-property. [ABSTRACT FROM AUTHOR]

Let A act on a group G. A is said to act p-cyclically on G if A N p A p - 1 acts stably on G / O p ′ (G) , and A is said to act cyclically on G if A acts p-cyclically on G for all primes p. In this paper, the actions of a group A on a p-group or p-soluble group G are investigated and some criteria of an action to be cyclic or p-cyclic are obtained. [ABSTRACT FROM AUTHOR]

Let P be the set of all prime numbers. A subgroup H of a finite group G is said to be P -subnormal in G if there exists a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n - 1 ⊆ H n = G such that either H i - 1 is normal in H i or | H i : H i - 1 | is a prime number for every i = 1 , 2 , … , n . A subgroup H of G is called a TI -subgroup if every pair of distinct conjugates of H has trivial intersection. The aim of this paper is to give a complete description of all finite groups in which every non- P -subnormal subgroup is a TI -subgroup. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π -property in G if there exists a chief series Γ G : 1 = G 0 < G 1 < ⋯ < G n = G of G such that for every G-chief factor G i / G i - 1 (1 ≤ i ≤ n) of Γ G , | G / G i - 1 : N G / G i - 1 (H G i - 1 / G i - 1 ∩ G i / G i - 1) | is a π (H G i - 1 / G i - 1 ∩ G i / G i - 1) -number. In this paper, we study the influence of some subgroups of prime power order satisfying the partial Π -property on the structure of a finite group. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies the Π-property in G if for any chief factor L/K of G, ∣G/K: NG/K(HK/K ∩ L/K)∣ is a π(HK/K∩L/K)-number. We study the influence of some p-subgroups of G satisfying the Π-property on the structure of G, and generalize some known results. [ABSTRACT FROM AUTHOR]

Abstract–In this paper, for every partition σ of the set of all primes, we obtain a complete classification of finite groups in which every subgroup is a σ-subnormal subgroup or a TI-subgroup. [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]

Let G be a finite group and σ = {σi ∣ i ⋃ I} be a partition of the set of all primes ℙ, that is, ℙ = ∪i∈Iσi and σi ∩ σj = ∅ for all i ≠ j. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of ℋ is a Hall σi-subgroup of G for some i and ℋ contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A group G is said to be a CLTσ-group if G has a complete Hall σ-set {H1, ..., Ht} such that for all subgroups Ai ≤ Hi, G has a subgroup of order ∣A1∣ ... ∣At∣. A group G is a generalized CLTσ-group, if G has a complete Hall σ-set {H1, ..., Ht} such that for all subgroups Ai ≤ Hi, G has a subgroup A of order ∣A1∣ ... ∣At∣ such that A satisfies some σ-normality condition. In this paper, we study the generalized CLTσ-groups which are characterized completely under some conditions. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P , that is, P = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j . The natural numbers n and m are called σ-coprime if σ (n) ∩ σ (m) = ∅ . The group G is said to be: σ-primary if G is a σi-group for some i ∈ I ; σ-soluble if either G = 1 or every chief factor of G is σ-primary. A subgroup H of G is called σ-subnormal in G if there is a subgroup chain H = H 0 ≤ H 1 ≤ ⋯ ≤ H t = G such that either H i − 1 is normal in Hi or H i / (H i − 1) H i is σ-primary for all i = 1 , ... , t . In this paper, we show that G is σ-soluble provided G satisfies the following conditions: (1) G = A 1 A 2 = A 1 A 3 = A 2 A 3 , where A1, A2, A3 are all σ-soluble; (2) the three indices | G : N G (A 1 N σ ) | , | G : N G (A 2 N σ ) | , | G : N G (A 3 N σ ) | are pairwise σ-coprime. And we prove that if G is σ-soluble and A is a subgroup of G, then A is σ-subnormal in G if and only if | A B | σ i divides | G | σ i for every Hall σi-subgroup B of G and all σ i ∈ σ (G) . We also state and prove a σ-nilpotency criterion for G and a characterization of the σ-Fitting subgroup of G, which are related to this observation. [ABSTRACT FROM AUTHOR]

Let X be a class of finite groups closed under subgroups, homomorphic images, and extensions. We study the question which goes back to the lectures of H. Wielandt in 1963–1964: Given an X -subgroup K and a maximal X -subgroup H, is it possible to detect embeddability of K in H (up to conjugacy) by their projections into the factors of a fixed subnormal series? On the one hand, we construct examples where K has the same projections as some subgroup of H but is not conjugate to any subgroup of H. On the other hand, we prove that if K normalizes the projections of a subgroup H, then K is conjugate to a subgroup of H even in the more general case when H is a submaximal X -subgroup. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies Φ - Π -property in G if for any non-Frattini chief factor L/K of G, | G / K : N G / K (H K / K ∩ L / K) | is a π (H K / K ∩ L / K) -number. In this paper, we study the influence of some subgroups of G satisfying Φ - Π -property on the structure of G, and obtain some interesting results. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called weakly s-semipermutable in G if G has a subnormal subgroup T and an s-semipermutable subgroup HssG contained in H such that G = HT and H ∩ T ≤ H ssG . In this paper, we present several new criteria on p-nilpotency of finite groups with a small quantity of weakly s-semipermutable maximal subgroups of Sylow p-subgroups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. We call that H has IC-property if the intersection of H and [H, G] satisfy some conditions. In this paper, we investigate the structure of finite group G under the assumption that some primary subgroups of G have IC-property and obtain some new results. [ABSTRACT FROM AUTHOR]

If σ = { σ i : i ∈ I } is a partition of the set P of all prime numbers, a subgroup H of a finite group G is said to be σ -subnormal in G if H can be joined to G by means of a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G such that either H i - 1 normal in H i or H i / Core H i (H i - 1) is a σ j -group for some j ∈ I , for every i = 1 , ... , n . If σ = { { 2 } , { 3 } , { 5 } ,... } is the minimal partition, then the σ -subnormality reduces to the classical subgroup embedding property of subnormality. A finite group X is said to be a Schmidt group if X is not nilpotent and every proper subgroup of X is nilpotent. Every non-nilpotent finite group G has Schmidt subgroups and a detailed knowledge of their embedding in G can provide a deep insight into its structure. In this paper, a complete description of a finite group with σ -subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba. [ABSTRACT FROM AUTHOR]

All groups considered are finite. Let F be a formation. A subgroup H of a group G is called K- F -subnormal in G if there exists a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G , with H i - 1 normal in H i or H i / Core H i (H i - 1) ∈ F for every 1 ≤ i ≤ n . If F = N , the formation of all nilpotent groups, the K- N -subnormal subgroups of a group G are exactly the subnormal subgroups of G. The aim of this paper is to prove the following theorem: if F is a subgroup-closed saturated lattice formation, then a subgroup H of a group G is K- F -subnormal in G if and only if H is K- F -subnormal in < H , x > for all x ∈ G . Some earlier results are consequence of this theorem. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P , that is, P = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j. A chief factor H/K of G is said to be σ-central in G if the semidirect product (H / K) ⋊ (G / C G (H / K)) is a σi-group for some i ∈ I. The group G is said to be σ-nilpotent if either G = 1 or every chief factor of G is σ-central in G. In this article, we study the properties and structures of a finite group G = AB, factorized by two σ-nilpotent subgroups A and B. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A finite group G is called a σ-soluble group if every chief factor of G is a σi-group for some σ i ∈ σ . In this note, we investigate how maximal subgroups of a σ-soluble group G are embedded in G. Some criteria for σ-solubility are obtained, which generalize some classical results. [ABSTRACT FROM AUTHOR]

F.G. Russo asked whether it is possible, or not, to find relations between the connected components of the prime graph and the Sylow graph of a finite group. We proved that either the prime graph or the Sylow graph of a finite group is connected. [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called σ -subnormal in G if there is a chain of subgroups X = X 0 ⊆ X 1 ⊆ ⋯ ⊆ X n = G where for every j = 1 , ⋯ , n the subgroup X j - 1 is normal in X j or X j / C o r e X j (X j - 1) is a σ i -group for some i ∈ I . In the special case that σ is the partition of P into sets containing exactly one prime each, the σ -subnormality reduces to the familiar case of subnormality. A finite group G is σ -complete if G possesses at least one Hall σ i -subgroup for every i ∈ I , and a subgroup H of G is said to be σ i -subnormal in G if H ∩ S is a Hall σ i -subgroup of H for any Hall σ i -subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel–Wielandt σ -problem: Is it true that a subgroupHof a σ -complete groupGis σ -subnormal inG if His σ i -subnormal in Gfor all i ∈ I ? The main goal of this paper is to solve the Kegel–Wielandt σ -problem for binary partitions. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be an I C Φ -subgroup of G if H ∩ [ H , G ] ≤ Φ (H) . We prove the p-nilpotency of a finite group G under the assumption that certain p-subgroups of G are I C Φ -subgroups of G. Our main result generalizes and extends some recent work of Gao and Li [1]. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P and G be a finite group. A group is said to be σ -primary if it is a finite σ i -group for some i. A subgroup A of G is said to be σ -subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G such that either A i - 1 ⊴ A i or A i / (A i - 1) A i is σ -primary for all i = 1 , ... , t . A subgroup S of G is m- σ -permutable in G if S = ⟨ M , B ⟩ for some modular subgroup M and σ -permutable subgroup B of G. We say that a subgroup H of G is m- σ -embedded in G if there exist an m- σ -permutable subgroup S and a σ -subnormal subgroup T of G such that H G = H T and H ∩ T ≤ S ≤ H , where H G = ⟨ H x | x ∈ G ⟩ is the normal closure of H in G. In this paper, we study the properties of m- σ -embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Throughout this article, all groups are finite. Let F be a formation of groups. A subgroup A of a finite group G is said to be K- F -subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i − 1 ⊴ A i or A i / (A i − 1) A i ∈ F for all i = 1 , ... , n . The formation F is called K-lattice if in every finite group G the set L K F (G) , of all K- F -subnormal subgroups of G, is a sublattice of the lattice L (G) of all subgroups of G. In this article, we prove that if F is a hereditary K-lattice saturated formation containing all nilpotent groups and G is a finite group such that every Schmidt subgroup of G is either K- F -subnormal or modular in G, then the derived subgroup G ′ of G belongs to F. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes ℙ , that is, ℙ = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i and H contains exactly one Hall σi-subgroup of G for every σ i ∈ σ (G). A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G such that either A i − 1 is normal in Ai or A i / (A i − 1) A i is σ-primary for all i = 1 , 2 , ... , t ; σ-abnormal in G if K / L K is not σ-primary whenever A ≤ L < K ≤ G and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set { H 1 , H 2 , ... , H t } such that for any A i ≤ H i , G has a subgroup A of order | A 1 | | A 2 | ⋯ | A t | and A is σ-abnormal or σ-subnormal in G. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let P be a Sylow p-subgroup of G. Under the hypothesis that NG(P) is p-nilpotent, we provide some conditions to give a p-nilpotency criterion of finite groups by Engel condition, which improves some recent results. [ABSTRACT FROM AUTHOR]

A subgroup A of a finite group G is called a local covering subgroup of G if AG = AB for all maximal G-invariant subgroup B of AG = 〈Ag, g ∊ G〉. Let p be a prime and d be a positive integer. Assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd = 2 and a Sylow 2-subgroup of G is nonabelian, of G are local covering subgroups. Then G is p-supersolvable whenever pd = p or p d ≤ | G | p or pd ≤ ∣Op'p(G)∣p/p. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called an I C Φ -subgroup of G if H ∩ [ H , G ] ≤ Φ (H) . We investigate the structure of a finite group G under the assumption that some subgroups of G are I C Φ -subgroups of G, and some results of p-nilpotency and supersolvability of finite groups are obtained. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We denote by N i l G (x) the set of elements y ∈ G such that ⟨ x , y ⟩ is a nilpotent subgroup and by ν 1 (G) and ν (G) the probability that two randomly chosen elements of G respectively generate an abelian subgroup and a nilpotent subgroup. A group G is called an N -group if N i l G (x) is a group for all x ∈ G . It is proved that G is supersolvable if there exists a normal subgroup H such that either ν 1 (H) > 1 3 | G : H | or G is an N -group and ν (H) > 1 3 | G : H | . [ABSTRACT FROM AUTHOR]

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|. In this paper, we prove a lower bound on |m(G)| when G is not nilpotent, that is, | m (G) | ≥ | π (G) | + p , where p ∈ π (G) is the smallest prime that divides |G| such that the Sylow p- subgroup of G is not normal in G. [ABSTRACT FROM AUTHOR]

Let G be a finite group, σ = { σ i | i ∈ I } a partition of the set of all primes P and σ (G) = { σ i | σ i ∩ π (| G |) ≠ ∅ }. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i ∈ I and H contains exactly one Hall σi-subgroup of G for every σ i ∈ σ (G). G is said to be σ-full if G possesses a complete Hall σ-set. We say a subgroup H of G is sσ-quasinormal (supplement-σ-quasinormal) in G if there exists a σ-full subgroup T of G such that G = HT and H permutes with every Hall σi-subgroup of T for all σ i ∈ σ (T). In this article, we obtain some results about the sσ-quasinormal subgroups and use them to determine the structure of finite groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set P of all primes, that is, P = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j. Let G be a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σ i ∈ σ (G). G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G is said to be σ-dispersive provided G has a normal series 1 = G 1 < G 2 < ⋯ < G t < G t + 1 = G and a complete Hall σ-set { H 1 , H 2 , ... , H t } such that G i H i = G i + 1 for all i = 1 , 2 , ... t. In this article, we give a characterization of σ-dispersive group, which gives a positive answer to an open problem of Skiba. [ABSTRACT FROM AUTHOR]

Throughout this article, all groups are finite and G is a group. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P. Then σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } ; σ + (G) = { σ i | G has a chief factor H/K such that σ (H / K) = { σ i } }. The group G is said to be: σ-primary if G is a σi-group for some i; σ-soluble if every chief factor of G is σ-primary. The symbol R σ (G) denotes the product of all normal σ-soluble subgroups of G. The chief factor H/K of G is said to be: σ-central (in G) if (H / K) ⋊ (G / C G (H / K)) is σ-primary; a σi-factor if H/K is a σi-group. We say that G is: σ-nilpotent if every chief factor of G is σ-central; generalized { σ i } -nilpotent if every chief σi-factor of G is σ-central. The symbol F { g σ i } (G) denotes the product of all normal generalized { σ i } -nilpotent subgroups of G. We call any function f of the form f : σ ∪ { ∅ } → { formations of groups } , where f (∅) ≠ ∅ , a generalized formation σ-function and we put B L F σ (f) = (G | G / R σ (G) ∈ f (∅) and G / F { g σ i } (G) ∈ f (σ i) for all σ i ∈ σ + (G)). If for some generalized formation σ-function f we have F = B L F σ (f) , then we say that the class F is Baer-σ-local and f is a generalized σ-local definitionof F. In this article, we describe basic properties, examples, and some applications of Baer-σ-local formations. In particular, we prove that the set of all Baer-σ-local formations containing all nilpotent groups forms a subsemigroup of the semigroup of all formations G G (Theorem 1.13) and the set of all Baer-σ-local formations containing all σ-nilpotent groups is a right ideal in G G (Theorem 1.16). [ABSTRACT FROM AUTHOR]