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Abstract–In this paper, based on the classifications of Kn-simple groups for n ≤ 6 and under the case that every maximal subgroup has a π-nilpotent trace, we obtained the necessary and sufficient conditions on solvable groups and proved that two unsolvable problems involving groups, rings and fields are true. [ABSTRACT FROM AUTHOR]

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]

AbstractLet G be a finite group. We write gsn(G) for the geometric mean Sylow number of G. The Fitting subgroup of G is denoted by F(G). It was recently proved that if gsn(G) < , then G is solvable. In this paper, we extend the study to nonsolvable groups. We prove that if G is a finite nonsolvable group, then the following holds:(a) if gsn(G) < and gsn(G) ≠ , then G/F(G) ≅ A5;(b) if gsn(G) = , then G/N ≅ A5, where N is the largest normal solvable subgroup of G. [ABSTRACT FROM AUTHOR]

We study the Wamsley group ⟨ X , Y , Z | X Z = X α , Z Y = Y β , Z γ = [ X , Y ] ⟩ and its Sylow subgroups, where α γ ≠ 1 ≠ β γ and γ > 0 , obtaining the sharpest results when α = β . [ABSTRACT FROM AUTHOR]

Let G be a finite p-separable group, for some fixed prime p. Let Γ p (G) be the common divisor graph built on the set of non-central conjugacy classes of p-regular elements of G: this is the graph whose vertices are the conjugacy classes of those non-central elements of G such that p does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in Γ p (G) between a vertex with maximal cardinality and any other vertex, and to study the p-structure of G when Γ p (G) has diameter three. [ABSTRACT FROM AUTHOR]

Given positive integers p and m, where p is assumed to be an odd prime, we determine the automorphism groups of p-groups J, H, and K of orders p 7 m , p 6 m , and p 5 m , and nilpotency classes 5, 4, and 3, respectively, all of which arise naturally from the Macdonald group 〈 x , y | x [ x , y ] = x 1 + p m ℓ , y [ y , x ] = y 1 + p m ℓ 〉 , where ℓ ∈ Z is relatively prime to p. [ABSTRACT FROM AUTHOR]

Let G be a finite group and N Ω (G) be the intersection of the normalizers of all subgroups belonging to the set Ω (G) , where Ω (G) is a set of all subgroups of G which have some theoretical group property. In this paper, we show that N Ω (G) = Z ∞ (G) if Ω (G) is one of the following: (i) the set of all self-normalizing subgroups of G; (ii) the set of all subgroups of G satisfying the subnormalizer condition in G; (iii) the set of all pronormal subgroups of G; (iv) the set of all weakly normal subgroups of G; (v) the set of all NE-subgroups of G. [ABSTRACT FROM AUTHOR]

AbstractLet σ={σi⏧i∈I} be some partition of the set P of all primes. A group G is said to be σ-supersoluble if all its chief factors below the σ-nilpotent residual GNσ are cyclic. In this paper, we give a classification on minimal non-σ-supersoluble groups. [ABSTRACT FROM AUTHOR]

AbstractWe consider complex characters of a p-group P, which are invariant under a fusion system F on P. Extending a theorem of Bárcenas–Cantarero to non-saturated fusion systems, we show that the number of indecomposable F -invariant characters of P is greater or equal than the number of F -conjugacy classes of P. We further prove that these two quantities coincide whenever F is realized by a p-solvable group. On the other hand, we observe that this is false for constrained fusion systems in general. Finally, we construct a saturated fusion system with an indecomposable F -invariant character, which is not a summand of the regular character of P. This disproves a recent conjecture of Cantarero–Combariza. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give a characterization of finite non-solvable groups which act flag-regularly on maps with negative square-free Euler characteristic. We also construct some interesting families of regular maps. [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]

We introduce two invariants for finite groups, and use each to characterize A5. [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]

We study the group structure in terms of the number of Sylow p-subgroups, which is denoted by np(G). The first part is on the group structure of finite group G such that np(G) = np(G/N), where N is a normal subgroup of G. The second part is on the average Sylow number asn(G) and we prove that if G is a finite nonsolvable group with asn(G) < 39/4 and asn(G) ≠ 29/4, then G/F(G) ≌ A5, where F(G) is the Fitting subgroup of G. In the third part, we study the nonsolvable group with small sum of Sylow numbers. [ABSTRACT FROM AUTHOR]

Given two subgroups H, K of a compact group G, the probability that a random element of H commutes with a random element of K is denoted by Pr (H , K) . We show that if G is a profinite group containing a Sylow 2-subgroup P, a Sylow 3-subgroup Q 3 and a Sylow 5-subgroup Q 5 such that Pr (P , Q 3) and Pr (P , Q 5) are both positive, then G is virtually prosoluble (Theorem 1). Furthermore, if G is a prosoluble group in which for every subset π ⊆ π (G) there is a Hall π -subgroup H π and a Hall π ′ -subgroup H π ′ such that Pr (H π , H π ′) > 0 , then G is virtually pronilpotent (Theorem 2). [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]

A finite group G is called an SNS-group if every subgroup of G is either subnormal in G or self-normalizing. In this article, we investigate the structure of finite groups which are not SNS-groups but all of whose maximal subgroups of even order are SNS-groups. In addition, we determine the minimal simple groups all of whose second maximal subgroups (respectively, second maximal subgroups of even order) are SNS-groups. [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]

We examine the p-nilpotency and supersolvablitiy of a finite group under the assumption that certain subgroups of prime power order are S ∗ - embedded in the group itself. In particular, we extend and generalize the recent results of Li (Commun. Algebra 50(4):1585–1594, https://doi.org/10.1080/00927872.2021.1986056, 2022) and the related results in the literature. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called a TI-subgroup of G if H ∩ H x = 1 or H for all x ∈ G . G N = ⋂ { N ⊴ G | G / N is nilpotent } is called the nilpotent residual of G. In this paper, we show that a non-abelian finite simple group the nilpotent residuals of all of its subgroups are TI-subgroups is isomorphic to either P S L (2 , 2 p) for some prime p, to PSL(2, 7) or to the Suzuki group S z (2 p) for some odd prime p. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group. We say that a subgroup H of G is a CSS-subgroup in G, if there exists a normal subgroup K of G such that G=HK and H∩K is SS-quasinormal in G. In this paper, we study the p-nilpotency and supersolvability of G under the assumption that every member of a small subset of the set of all maximal subgroups of the Sylow p-subgroups P of G with certain property is a CSS-subgroup in G. Using this criterion, we obtain new results which generalize and unify recent results in the literature. [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 be a finite group and G be a split extension of A by B, that is, G is a semidirect product: G = A ⋊ B , where A and B are subgroups of G. Under the condition that B permutes with every maximal subgroup of Sylow subgroups of A, every maximal subgroup of A or every nontrivial normal subgroup of A, we prove that the supersolvable residual of G is the product of the supersolvable residuals of A and B. [ABSTRACT FROM AUTHOR]

It is well known that, up to isomorphism, there is a unique simple group of order 168. The projective special linear groups PSL(2, 7) and PSL(3, 2) are concrete examples of such groups, and are therefore isomorphic. This isomorphism is not obvious, but it has been clarified from several viewpoints. Here we use purely group-theoretic methods—the Sylow theorems, Poincaré's theorem, and the orbit-stabilizer theorem—to show that any simple group of order 168 has a natural action on the projective line over F 7 , so it is isomorphic to PSL(2, 7). We discuss an analogous action on the projective plane over F 2 described by Smith and Tabachnikova. Both geometries are constructed using conjugacy classes of maximal subgroups. This gives another proof of the uniqueness theorem and may dispel some of the air of mystery that often surrounds the isomorphism between PSL(2, 7) and PSL(3, 2). [ABSTRACT FROM AUTHOR]

Let F be a saturated fusion system over a p-group S and let E be a subsystem of F with Aut E (S) = Aut F (S) . We prove that the fusion system F is generated by all E -morphisms and all F -automorphisms of some domestic intersections (called F ∖ E -domestic intersections in the paper) of F . We also get some results about control of fusion in finite groups as applications. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let π be a set of primes. The aim of this paper is to obtain some results concerning how much information about the π -structure of G can be gathered from the knowledge of the sizes of conjugacy classes of its π -elements and of their multiplicities. Among other results, we prove that this multiset of class sizes determines whether G has a hypercentral Hall π -subgroup. [ABSTRACT FROM AUTHOR]

AbstractLet G and A be finite groups of relative coprime orders and A acts on G via automorphisms. In this paper, we investigate the p-nilpotency and solvability of finite groups in which every maximal A-invariant subgroup of a Sylow p-subgroup of G for some prime p is S-semipermutable. [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]

For a finite group G, let s(G) denote the number of subgroups of G and d(G) denote the number of divisors of |G|. In this paper, we prove that there does not exist a finite group G such that d (G) < s (G) < d (G) + p 1 , where p 1 is the smallest prime divisor of |G|, and determine the structure of finite groups G such that s (G) = d (G) + p 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 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]

The aim of this article is to study how the greatest common divisor of the degree and codegree of an irreducible character of a finite group influences its structure. We study a finite group GG with gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) a prime for almost all irreducible characters χ\chi of GG, and obtain the following two conclusions: (1)There does not exist any finite group GG such that gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ∈Irr(G)♯\chi \in {\rm{Irr}}{\left(G)}^{\sharp }, where Irr(G)♯{\rm{Irr}}{\left(G)}^{\sharp } is the set of non-principal irreducible characters of GG.(2)Let GG be a finite group, if gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ∈Irr(G)\Lin(G)\chi \left\in {\rm{Irr}}\left(G)\backslash {\rm{Lin}}\left(G), then GG is solvable, where Lin(G){\rm{Lin}}\left(G) is the set of all linear irreducible characters of GG.

We consider 9 infinite families of finite p-groups, for p a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified. [ABSTRACT FROM AUTHOR]

Let G be a finite group. In this paper, we obtain some new conditions for supersolvability of G with three supersolvable subgroups of pairwise relatively prime indices in G. Our main results improve and extend some known results in the literature. [ABSTRACT FROM AUTHOR]

For a prime p, we show that uniqueness of factorization into irreducible Σ p 2 -invariant representations of Z / p ≀ Z / p holds if and only if p = 2 . We also show nonuniqueness of factorization for Σ 8 -invariant representations of D 8 ≀ Z / 2 . The representation ring of Σ p 2 -invariant representations of Z / p ≀ Z / p is determined completely when p equals two or three. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the structure of groups with the property of the core relation between some second maximal subgroups and corresponding maximal subgroups, and obtained some new results about finite groups. ARTICLE HISTORY Received 7 September 2023 Revised 13 December 2023 Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in { 1 , 2 , ⋯ , n - 1 } . Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group G is said to admit a uniform group factorization if there exist subgroups H 1 , H 2 , ... , H k such that G = H 1 H 2 ⋯ H k and the number of ways to represent any element g ∈ G as g = h 1 h 2 ⋯ h k ( h i ∈ H i ) does not depend on the choice of g. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations. [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 G be a finite group and p be a prime. We define N N p ∗ (G) to be the intersection of the normalizers of the p-nilpotent residuals of all two-generator subgroups of G whose p-nilpotent residuals are nilpotent. We show that N N p (G) = N N p ∗ (G) . Using the method in the present paper, we will be able to give an affirmative answer to an open problem in Shen et al. (Mediterr J Math 19:191, 2022), which also indicates that similar conclusions hold for many formations. It is also proved that G = N N p (G) if and only if every three-generator subgroup H of G satisfies H = N N p (H) . To this end, we introduce and investigate the IO- N N p -groups, i.e., the groups G such that G ≠ N N p (G) , but each proper subgroup and each proper quotient of G equals its p-nilpotent norm. Moreover, new results in terms of the p-nilpotent norm and the p-nilpotent hypernorm N ∞ N p (G) are given. [ABSTRACT FROM AUTHOR]

Let X be a class of groups. A group G is called a X -critical group if G ∉ X whereas every proper subgroup of G is in X . We call G a pd-group if |G| is divisible by a prime p. A group G is called a X -semicritical group with respect to a prime p if G ∉ X , but every proper pd-subgroup of G is in X . Let U be a class of supersolvable groups. In this paper, we mainly study the U -semicritical groups with respect to 2. Furthermore, we describe the non-solvable groups whose every 2d-maximal subgroup is U -semicritical groups with respect to 2. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of a group G is called pronormal in G if the subgroups H and H g are conjugate in ⟨ H , H g ⟩ for each g ∈ G . A group G is said to be a PRN-group if every minimal subgroup of G or order 4 is pronormal in G. In this paper, we characterize groups G such that G is a non-PRN-group of even order in which every maximal subgroup of even order is a PRN-group, and come to that such groups are solvable, have orders divisible by at most 3 distinct primes. And some additional structural details are provided. [ABSTRACT FROM AUTHOR]

Let G be a finite group and δ (G) = 1 | G | ∑ H ≤ G { | H | | H is non-cyclic}. In this paper, we show that some arithmetical conditions of δ (G) influence the structure of G. Firstly, we prove that if δ (G) < 13 3 , then G is solvable. Secondly, we determine the structure of finite groups with δ (G) ≤ 2 . Moreover, we prove that if δ (G) < 1 + 4 | G | , then G is supersolvable, and we also determine the structure of finite groups G with δ (G) = 1 + 4 | G | . Finally, we show that δ (G) < c does not imply the supersolvability of G for any constant c ∈ (1 , ∞) . [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]