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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]

A finite group is called a weakly Dedekind group if all its noncyclic subgroups are normal. In this paper, we determine the complete classification of weakly Dedekind groups. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian group and Z(G) be its center. We associate a commuting graph Γ (G) to G, whose vertex set is G \ Z (G) and two distinct vertices are adjacent if they commute. In this paper we prove that the set of all non-abelian groups whose commuting graph has maximum vertex degree bounded above by a fixed k ∈ N is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a group G. The permutizer P G (H) is the subgroup generated by all cyclic subgroups of G which permute with H. A subgroup H of a group G is strongly permuteral in G if P U (H) = U for every subgroup U of G, such that H ≤ U ≤ G . We investigate groups with P -subnormal or strongly permuteral Sylow subgroups. Moreover, we prove that groups with all strongly permuteral primary cyclic subgroups are supersoluble. [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 G be a finite group with the identity element e. The proper order graph of G, denoted by â* (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of â*(G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups.

Given a finite group G, we introduce "encoding pairs," which are a pair of G-modules M and M ′ equipped with a shifted natural isomorphism between the cohomological functors H • (G , Hom Z (M , −)) and H • (G , Hom Z (M ′ , −)) . Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere. [ABSTRACT FROM AUTHOR]

In this paper, the norm Li(G) of the lower central series in a finite group G is introduced, which unifies the norm of derived subgroups and nilpotent residuals. Some propositions of Li(G) are obtained, and some related subgroups as well as their equivalent propositions can also be found. [ABSTRACT FROM AUTHOR]

Let G be a Frobenius group with the Frobenius kernel K. Suppose that G contains a nontrival subgroup D ⊆ K such that the normalizer N G (D) ⊆ K . When D is not a 2-group, Flavell proved, without using character theory, that K is a subgroup of G. Based on this result, we further prove that K is a subgroup when D is a 2-group. [ABSTRACT FROM AUTHOR]

For an irreducible complex character χ of a finite group G, its codegree is defined as |G:kerχ|/χ(1)$|G:\ker \chi |/\chi (1)$. The main focus of this paper is to provide a classification of finite groups where distinct nonlinear irreducible characters have distinct codegrees. [ABSTRACT FROM AUTHOR]

A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau)$ -regular if R induces a $\kappa $ -regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau)$ -regular set of some Cayley graph on a finite group G , then R is called a $(\kappa ,\tau)$ -regular set of G. Let H be a nontrivial normal subgroup of G , and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$ , $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau)$ -regular set of G ; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau)$ -regular set of G if and only if it is a $(0,1)$ -regular set of G. [ABSTRACT FROM AUTHOR]

Let G be a finite group and Irr(G) be the set of irreducible characters of G. The codegree of an irreducible character χ of the group G is defined as cod(χ) = |G : ker(χ)|/χ(1). In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the p-parts of the codegrees and character degrees. [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]

A combinatorial block design D is called f -pyramidal if there exists a subgroup G of Aut (D) fixing f points and acting regularly on the other points. If this happens, we say that the design is f -pyramidal under G. In case D is a 3-pyramidal Kirkman triple system, it is known that such a group G has precisely 3 involutions, all conjugate to each other. In this paper, we obtain a classification of the groups with this property. [ABSTRACT FROM AUTHOR]

Let be a partition of the set of all primes, and let be a finite group. The group is said to be -primary if is a -group for some and -complete if has a Hall -subgroup for each . A subgroup of is (i) -subnormal in if it has a subgroup series such that either or is -primary for each ; (ii) modular in if (1) for all such that and (2) for all such that ; (iii) -quasinormal in if is -subnormal and modular in . Finite solvable groups in which the -quasinormality of subgroups is a transitive relation are described. Some known results are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Brauer's height zero conjecture, Robinson's conjecture, and Olsson's conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results. [ABSTRACT FROM AUTHOR]

A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G = AT and for any X ≤ A and for any Y ≤ T there exists an element u ∈ ⟨X, Y⟩ such that XYu ≤ G. The notation H ≤ G means that H is a subgroup of a group G. In this paper, we obtained the estimations of numerical invariants(the derived length, the nilpotent length, the π-length, the nilpotent π-length) of G = AB in terms of invariants of tcc-subgroups A and B. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let V be a finite completely reducible faithful G-module, possibly of mixed characteristic. The second author and Yong Yang proved in previous work that | G : G ′ | ≤ M , where M is the largest orbit size of G on V. They also showed that if G is a nonabelian group and | G : G ′ | = M , then G is nilpotent having at least two orbits of maximal size M on V. In this paper, we deal with the latter case, where G is nonabelian and | G : G ′ | = M , and we classify all linear group actions in which G has at most three orbits of size M on V. [ABSTRACT FROM AUTHOR]

Let σ = { π i ∣ i ∈ I } be a partition of the set of all primes. We characterize the class of all σ -nilpotent groups as a hereditary formation F that contains every group G all whose Sylow subgroups are K- F -subnormal in their product with the generalized Fitting subgroup F ∗ (G) . [ABSTRACT FROM AUTHOR]

Let S be a subset of a finite group Γ. The Cayley signed graph, denoted by Σ = CayΣ(S, σ) has vertex set Γ and two distinct vertices x, y ∈ Γ are joined by an edge from x to y if and only if there exists s ∈ S such that x = sy, where Σ is a signed graph whose underlying graph is Γ and σ : E(Γ) → {+, -} is a function defined as ... In this manuscript, we have characterized the Cayley set and generating sets for which Cayley signed graphs are canonically consistent. [ABSTRACT FROM AUTHOR]

Earlier, the author described up to conjugacy all pairs 퐴 퐵 of nilpotent subgroups of a finite group 퐺 with socle subscript 퐿 2 푞 for which 퐴 superscript 퐵 푔 1 for any element of 퐺 . A similar description was obtained by the author later for primary subgroups 퐴 and 퐵 of a finite group 퐺 with socle subscript 퐿 푛 superscript 2 푚 . In this paper, we describe up to conjugacy all pairs 퐴 퐵 of nilpotent subgroups of a finite group 퐺 with simple socle from the "Atlas of Finite Groups" for which 퐴 superscript 퐵 푔 1 for any element 푔 of 퐺 . The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the "Kourovka Notebook") that a finite simple nonabelian group 퐺 for any nilpotent subgroups 푁 contains an element 푔 such that 푁 superscript 푁 푔 1 . [ABSTRACT FROM AUTHOR]

Let 픉 be a formation, and let 퐺 be a finite group. A subgroup 퐻 of 퐺 is called K 픉 -subnormal (submodular) in 퐺 if there is a subgroup chain 퐻 subscript 퐻 0 subscript 퐻 1 ... subscript 퐻 푛 1 subscript 퐻 푛 퐺 such that, for every 푖 either subscript 퐻 푖 is normal in subscript 퐻 푖 1 or superscript subscript 퐻 푖 1 픉 subscript 퐻 푖 ( subscript 퐻 푖 is a modular subgroup of subscript 퐻 푖 1 , respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is K subscript 픘 1 -subnormal. Here subscript 픘 1 is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation 픉 , every solvable K 픉 -subnormal subgroup of a group 퐺 is contained in the solvable radical of 퐺 . We also obtain a series of applications of these results to the investigation of groups factorized by K 픉 -subnormal and submodular subgroups. [ABSTRACT FROM AUTHOR]

For an arbitrary partition 휎 of the set ℙ of all primes, a sufficient condition for the 휎 -subnormality of a subgroup of a finite group is given. It is proved that the Kegel–Wielandt 휎 -problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1. [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]

Let G be a finite group and χ be an irreducible character of G, the number cod (χ) = | G : ker (χ) | / χ (1) is called the codegree of χ. Also, cod(G) = {cod(χ) | χ ∈Irr(G)}. For d ∈cod(G), the multiplicity of d in G, denoted by m G ′ (d) , is the number of irreducible characters of G having codegree d. A finite group G is called a T k ′ -group for some integer k ≥ 1, if there exists d0 ∈cod(G) such that m G ′ (d 0) = k and for every d ∈cod(G) −{d0}, we have m G ′ (d) = 1 . In this note we characterize finite T k ′ -groups completely, where k ≥ 1 is an integer. [ABSTRACT FROM AUTHOR]

Let be a finite group and S be a non-empty subset of A Cayley graph of the group denoted by Cay(S) is defined as a simple graph that its vertices are the elements of and two vertices u and v are adjacent if uv1 2. The minimum edge dominating energy of Cayley graph Cay(; S) is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph Cay(; S). In this paper, we estimate the minimum edge dominating energy of the Cayley graphs for the finite group Sn. [ABSTRACT FROM AUTHOR]

Consider G to be a finite group and p to be a prime divisor of the order | G | in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ ∈ I r r (G) have a positive height, and let there be an anchor group A ϕ . We prove that if the normalizer N G (A ϕ) is p-constrained, then O p ´ (N G (A ϕ)) ≠ { 1 G } , where O p ´ (N G (A ϕ)) is the maximal normal p ´ subgroup of N G (A ϕ) . We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the 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]

Abstract Let G be a finite group and χ ∈ Irr ( G ) . The codegree of χ is defined as cod ( χ ) = | G : ker ( χ ) | / χ ( 1 ) and cod ( G ) = { cod ( χ ) | χ ∈ Irr ( G ) } is called the set of codegrees of G. In this paper, we show that the set of codegrees of U 4 ( 2 ) , S 4 ( q ) ( q = 2 f ≥ 4 ) , U 4 ( 3 ) , 2 F 4 ( 2 ) ′ , J 3 , G 2 ( 3 ) , A 9 , J 2 , PSL ( 4 , 3 ) , McL , S 4 ( 5 ) , G 2 ( 4 ) , HS , O ′ N and M 24 determines the group up to isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer. [ABSTRACT FROM AUTHOR]

For a nonempty set ω of primes, V. A. Vedernikov had constructed ω-fibred formations of groups via function methods. We study lattice properties of ω-fibred formations of finite groups with direction δ satisfying the condition δ0 ≤ δ. The lattice ωδFθ of all ω-fibred formations with direction δ and θ-valued ω-satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ωδF, ωδFτ, τωδF, ωδnF of ω-fibred formations of groups are shown to be algebraic. [ABSTRACT FROM AUTHOR]

In this paper, among other results, we give some sufficient conditions for every non-abelian subgroup of a group to be isoclinic with the group itself. It is also seen that under certain conditions, two groups have same number of element centralizers implies they are isoclinic. We prove that if G is any group having 4, 5, 7 or 9 element centralizers and H is any non-abelian subgroup of G, then | Cent (G) | = | Cent (H) | and G ′ ≅ H ′ ≅ C 2 , C 3 , C 5 or C 7 respectively. Furthermore, it is proved that if G is any group having n ∈ { 4 , 5 , 6 , 7 , 9 } element centralizers, then | G ′ | = n − 2 . Communicated by Mark Lewis [ABSTRACT FROM AUTHOR]

For a nontrivial finite group G, the intersection graph Γ (G) of G is the simple undirected graph whose vertices are the nontrivial proper subgroups of G and two vertices are joined by an edge if and only if they have a nontrivial intersection. In a finite simple graph Γ , the clique number of Γ is denoted by ω (Γ) . In this paper we show that if G is a finite group with ω (Γ (G)) < 13 , then G is solvable. As an application, we characterize all non-solvable groups G with ω (Γ (G)) = 13 . Moreover, we determine all finite groups G with ω (Γ (G)) ∈ { 2 , 3 , 4 } . [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P and G a finite group. Then G is said to be σ-full if G has a Hall σ i -subgroup for all i ∈ I. A subgroup A of G is said to be σ-permutable in G provided G is σ -full and A permutes with all Hall σ i -subgroups H of G (that is, A H = H A) for all i. We obtain a description of finite σ -full groups G with supersoluble Hall σ i -subgroups for all i ∈ I in which σ -permutability is a transitive relation on G. [ABSTRACT FROM AUTHOR]

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ 0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group. [ABSTRACT FROM AUTHOR]

Let 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 H G . In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power order are Hall normally embedded in G. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called F ∗ (G) -subnormal if it is subnormal in the product with the generalized Fitting subgroup F ∗ (G) of G. All saturated formations that contain every group G = A B where A and B are F ∗ (G) -subnormal -subgroups are described. Moreover, if such formation is also normally hereditary, then every group G = A B where A and B are F ∗ (G) -subnormal quasi- -subgroups is a quasi- -group. The connection of above-mentioned formations to the lattice formations, GWP-formations and formations with the Kegel property is found. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be a 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 non-identity member of ℋ is a Hall σ i -subgroup of G for some i ∈ I and ℋ contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ (G). A subgroup H of G is said to be: σ -permutable in G if G has a complete Hall σ -set ℋ such that H A g = A g H for all A ∈ ℋ and all g ∈ G ; σ -semipermutable in G if G has a complete Hall σ -set ℋ such that H A g = A g H for all g ∈ G and all σ i -group A ∈ ℋ with σ i ∈ σ (G) ∖ σ (H). We say that H is σ -semiembedded in G if there exists a σ -permutable subgroup T of G such that H T is a σ -permutable subgroup of G and H ∩ T ≤ H σ s G , where H σ s G denotes the subgroup of H which is generated by all σ -semipermutable subgroups of G contained in H. In this paper, we study the influence of σ -semiembedded subgroups on the structure of finite groups. Some known results are generalized and unified. [ABSTRACT FROM AUTHOR]

Recall that a subgroup A of a group G is called G-permutable in G if for every subgroup B of G there exists an element x ∈ G such that A and B x commute. The following question was posed in the Kourovka Notebook: is there an integer n such that for all m > n the alternating group A m has no non-trivial A m -permutable subgroups? We give a positive answer to this question. Moreover, in the case of prime p we prove that A p has no non-trivial A p -permutable subgroups except p = 5 . [ABSTRACT FROM AUTHOR]

Given a prime power p d where p is a prime and d is a positive integer, we aim to classify two classes of finite groups based on their complementary properties of certain p-subgroups. The first class of groups are those in which p 2 d divides the order of the groups and all subgroups of order p d are complemented. The second class of groups are those having a normal elementary abelian Sylow p-subgroup P, with p d less than | P | , and all subgroups of order p d are complemented. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G the number cod(χ) = |G: Ker(χ)|/χ(1) is called the codegree of χ. In this note, we verify the equivalent version of Huppert's conjecture for K3 -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]

Let G be a finite group. For an element x of G, xG denotes the conjugacy class of x in G. |xG| is called the size of the conjugacy class xG. Let N be a normal subgroup of G. For x ∈ N, we have xG ⊆ N and xG is called a G-class of the normal subgroup N. In this paper, we develop several results on the p -part of G-class sizes of a normal subgroup of a finite group G. [ABSTRACT FROM AUTHOR]

In this paper, the structure of a finite group 퐺 all of whose minimal subgroups are hereditarily 퐺 -permutable is studied. [ABSTRACT FROM AUTHOR]