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In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

Let o (G) be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between o (G) and o (H) , where H is a subgroup of G. Let N be the class of finite nilpotent groups and let L (G) be the subgroup lattice of G. In this paper, we show that the set { o (G) o (H) | G ∈ N , H ∈ L (G) } is dense in [ 0 , ∞) . Other density results are outlined throughout the paper. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we partially answer a question of Wujie Shi about the width of the order of the elements of a finite group. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group, and let p be a prime. In this paper, we obtain some conditions for G to be p-nilpotent or p-closed in terms of strongly monolithic characters. [ABSTRACT FROM AUTHOR]

AbstractThe aim of this paper is to investigate the influence of intrinsic properties of maximal subgroups on Coleman automorphisms of finite groups G. Let M be a maximal subgroup of G. It is proved that if M is simple then OutCol(G)=1 . It is also proved that either OutCol(G) is trivial or OutCol(G) is an abelian p-group for some prime p provided that M has a unique nontrivial proper normal subgroup. As an application, it is obtained that under some additional conditions the normalizer property holds for such groups. [ABSTRACT FROM AUTHOR]

Let F be a class of finite groups. A group G is called a minimal non- F -group or simply an F -critical group, if G is not in F but all proper subgroups of G are in F . Let Sch(G) denotes the set of all Schmidt subgroups of a group G (i.e. N -critical subgroups of G, where N is the class of all nilpotent groups), and Sch U = { G | Sch (G) ⊆ U } , where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class Sch U . In particular, we prove that Sch U is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every Sch U -critical group is a Schmidt group. In addition, we show that Sch U is locally defined by the formation function f such that f (p) = S { p } ∪ π (p − 1) for every prime p. Besides, we describe all Sch U -critical groups and all minimal non-supersoluble groups that belong to Sch U . [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]

AbstractLet G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈x〉∩〈y〉≠{e} or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension |G|−2. Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group with automorphism group Aut(G) and g∈G . For any subgroup H of Aut(G) , define a map ξH:G→C by ξH(g)=|{(x,α)∈G×H:[x,α]=g}| , where [x,α]:=x−1α(x) is called the autocommutator of x and α in G. In this paper, we show that ξH is a character of G when H is any subgroup of Aut(G) containing Inn(G) and H=Autcent(G) , where Inn(G) is the inner automorphism group of G and Autcent(G) is the group of all the central automorphisms of G. In these cases expressions for ξH are also obtained in terms of irreducible characters of G. Our results generalize a classical result of Frobenius regarding the number of solutions of the commutator equation in finite groups. As applications of our results, we compute g-autocommuting probabilities of some finite groups and obtained certain bounds for g-autocommuting probability of G in terms of commuting and autocommuting probabilities. [ABSTRACT FROM AUTHOR]

Let G(q) be a twisted or untwisted finite exceptional group of Lie type over GF (q) , either adjoint or simply connected, and let W be the Weyl group of G(q). This paper compiles an extensive data set on the maximal tori of G(q). A systematic labeling is given, via the corresponding conjugacy classes of W, together with a complete list of the structures of the maximal tori in terms of their torsion coefficient decomposition. Additionally the associated admissible graphs and the structure or shape of centralizers of elements in W are given as well as compatible computer files. Also various apparent inconsistencies in the current literature on simply connected groups are discussed and resolved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate whether the Uniqueness Theorem introduced by Feit and Thompson holds in finite simple groups of Lie type of Lie rank one. We also introduce and examine the Strong Uniqueness Theorem. We define the concept of containment minimality and prove that a non-Abelian finite simple group is minimal if and only if it is containment minimal. Finally, we inquire when a minimal non-soluble group has a faithful doubly transitive representation. [ABSTRACT FROM AUTHOR]

It is very difficult to determine the full automorphism group of compact Riemann and Klein surfaces. The automorphism groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 7 are known, but a lot of hard work is left to do. To amplify the knowledge in this field, in this paper the full automorphism groups of the surfaces of genus 8 are calculated. [ABSTRACT FROM AUTHOR]

AbstractLet G be an abelian group, and let F(G) be the free commutative monoid with basis G. For Ω⊂F(G), define the universal zero-sum invariant dΩ(G) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω. The invariant dΩ(G) unifies many classical zero-sum invariants. Let B(G) be the submonoid of F(G) consisting of all zero-sum sequences over G, and let A(G) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B(G), is denoted by ε. The well-known Davenport constant D(G) of the group G can be also represented as dB(G)∖{ε}(G) or dA(G)(G) in terms of the universal zero-sum invariant. Notice that A(G) is the unique minimal generating set of the monoid B(G) from the point of view of Algebra. Hence, it would be interesting to determine whether A(G) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A(G) such that dΩ(G)=D(G). Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant dΩ;Ψ(G) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant DΨ(G) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that DΨ(G)<∞ in terms of the weights set Ψ when |Ψ| is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant. [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, and g∈G. Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ(g)=0. Denote by Vo(G) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo(N)=Vo(G) is isomorphic to G. In this paper, we investigate the V-recognizability of E8(p), where p is a prime number. As an application, among the 610 primes p with p<10000 and p≡0,1,4 ( mod 5), we obtain that the method is always valid for confirming the V-recognizability of E8(p) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589. [ABSTRACT FROM AUTHOR]

For an irreducible character χ of a finite group G, the codegree of χ is defined as | G : ker (χ) | / χ (1) . In this paper, we characterize finite p-groups with three character codegrees. [ABSTRACT FROM AUTHOR]

Let p be a prime and F p be a finite field of p elements. Let F p G denote the group algebra of the finite p-group G over the field F p and V (F p G) denote the group of normalized units in F p G . Suppose that G is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p , n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V (F p G) p l and G p l , Ω l (V (F p G)) and Ω l (G) are given. Furthermore, there is a direct proof for V (F p G) p ∩ G = G p . [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 of H in G. In this paper, we investigate the structure of a finite group G under the assumption that certain 2-minimal subgroups and 2-maximal subgroups are Hall normally embedded in G, respectively. Some conditions for a finite group to be p-nilpotent and supersolvable are given. [ABSTRACT FROM AUTHOR]

A representation of a finite group G on a finite dimensional vector space V is called unisingular if every g ∈ G has 1 as an eigenvalue in its action on V. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over Q. We additionally identify new unisingular representations of the symmetric and alternating groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the isotropy group of non-simple derivations of the form x m y u D , where D is a simple derivation of K[x, y] and m, u are non-negative integers. For m , u ≥ 1 , we have proved that the isotropy group of x m y u D is isomorphic to the product of two finite cyclic groups. Further, we have proved that isotropy group of derivations x m D and y u D are finite, where m, u are positive integers. [ABSTRACT FROM AUTHOR]

For a positive integer l and a finite group G, let S n m l (G) = Σ χ ∈ Irr n m (G) χ l (1) Σ χ ∈ Irr n m (G) χ l − 1 (1) , where Irr n m (G) denotes the set of non-monomial characters of G. In this paper, we prove that if S n m l (G) < S n m l ( SL 2 (5)) , then G is solvable. In particular, we show that for a non-solvable group G, if S n m l (G) = S n m l ( SL 2 (5)) , then G = MC is a central product of two subgroups M and C such that M ≅ SL 2 (5) , Z (M) = M ∩ C and G/M is an M-group. Among the other results, we arrive at the solvability of G by assuming Σ χ ∈ Irr n m (G) χ (1) > 9 | G | / 29 . [ABSTRACT FROM AUTHOR]

Let σ = { σ i ∣ i ∈ I } be some partition of the set of all primes P , G a finite group, F a class of finite groups, σ (G) = { σ i ∣ σ i ∩ π (G) ≠ ∅ } , and σ (F) = ∪ G ∈ F σ (G). A function f of the form f : σ → { formationsofgroups } 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 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 class of groups F is called τ-closed if τ (G) ⊆ F for all G ∈ F . In this paper we prove that the lattice of all τ-closed n-multiply σ-local formations is an algebraic modular lattice of formations. [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]

Let G be a finite group and cd (G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group with socle H 0 = 2 G 2 (q) , where q = 3 f with f ⩾ 3 odd such that cd (G) = cd (H) , then G is non-solvable and the chief factor G ′ / M of G is isomorphic to H0. If, in particular, f is coprime to 3, then G ′ is isomorphic to H0 and G / Z (G) is isomorphic to H. [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 H G , where H G is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called an HNE2-group if all cyclic subgroups of order 2 and 4 of G are Hall normally embedded in G. In this paper, we prove that all HNE2-groups are 2-nilpotent. Furthermore, we also characterize the structure of finite group all of whose maximal subgroups are HNE2-groups. Finally, we determine finite non-solvable groups all of whose second maximal subgroups are HNE2-groups. [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 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]

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. In this paper, for all odd numbers k≥3, using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

AbstractLet σ={σi|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σi|σi∩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 σi∈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]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers s ≥ 2 and n ≥ 2 , with gcd (p , s) = 1 . In this paper, some relations between cs(G) and | Cent (G) | are established in the case where cs (G) = { 1 , p n , p n − 1 s } . Further, when p ∈ { 2 , 3 } , we determine the values of s and the structure of a finite group G such that cs (G) = { 1 , p n , p n − 1 s } . We also describe the structure of an AC-group G such that [ G : Z (G) ] = 3 n s and | Cent (G) | = 1 + ∑ i = 0 n 3 i , where gcd (3 , s) = 1 . [ABSTRACT FROM AUTHOR]

A way to construct the natural representation of the quantized affine algebra U v ( s l ̂ l) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for U q (s l N) were q is a root of unity to the action of U v ( s l ̂ l) as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional U q (g) -modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities. [ABSTRACT FROM AUTHOR]

AbstractFor a character χ of a finite group G, the number cod(χ)=[G:Kerχ]χ(1) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod(χ)≠cod(ϕ), the greatest common divisor of cod(χ) and cod(ϕ) is a prime-power, then either G is solvable or G/Sol(G) is isomorphic to one of the groups Alt5 or PSL2(8), where Sol(G) is the solvable radical of the group G. [ABSTRACT FROM AUTHOR]

AbstractLet G be a group and A, B, C be subgroups of G. In this paper, we prove that if G=AB∪BC, then G = AB or G = BC. But, in general, G=AB∪AC does not imply that G = AB or G = AC. We also prove that if G is a 2-decomposed group, and G is a Sylow tower group or G is nilpotent-by-nilpotent, then G=AB∪AC implies that G = AB or G = AC. Other interesting special cases are also discussed. [ABSTRACT FROM AUTHOR]

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 (K) of some finite regular Galois extension F / K (T) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group. [ABSTRACT FROM AUTHOR]

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p ′ -order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors. [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]

For a finite group G, and a subset S of G (possibly, contains the identity element), the bi-Cayley graph Bcay(G, S) of G with respect to S is defined as the undirected bipartite graph with vertex set G × { 0 , 1 } and edge set { { (g , 0) , (g s , 1) } | g ∈ G , s ∈ S } . In this paper, we give some sufficient conditions for the existence of hamiltonian cycle in Bcay (G = Z m ⋊ H , S) , where G = Z m ⋊ H is a semiproduct of Zm by a subgroup H of G. In addition, we introduce a new graph operation, which produces some classes of bi-Cayley graphs having hamiltonian cycle. [ABSTRACT FROM AUTHOR]

Let G be a finite π-separable group, where π is a set of primes, and let χ be an irreducible complex character that is a π-lift of some π-partial character of G. It was proved by Cossey and Lewis that all of the vertex pairs for χ are linear and conjugate in G if 2 ∈ π , but the result can fail for 2 ∉ π . In this paper we introduce the notion of the linear twisted vertices in the case where 2 ∉ π , and then establish the uniqueness for such vertices under the conditions that either χ is an N -lift for a π-chain N of G or it has a linear Navarro vertex, thus answering a question proposed by them. [ABSTRACT FROM AUTHOR]

Abstract–Given a finite group G, an element is called a vanishing element of G if it is a zero of an irreducible character of G; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is termed as a vanishing class if that element is a vanishing element. In this paper, we describe the finite groups whose vanishing class sizes are all powers of a prime p. [ABSTRACT FROM AUTHOR]

In this paper, we describe the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the non-abelian groups and then we investigate them in non-abelian finite groups in detail. Furthermore, we give the lengths of the periods of the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the polyhedral group (n , 2 , 2) with respect to the generating triple (a , b , c) as applications of the results produced. [ABSTRACT FROM AUTHOR]

In this paper, we study how certain arithmetical conditions on the lengths of conjugacy classes of primary elements and π-elements of a finite group G influence the structure of G, some new results are obtained and several well-known results in the literature are developed. In particular, the structure of a kind of outer p-supersolvable groups is described, some new conditions implying π-supersolvability are found, and the upper bound for the order of Sylow subgroups is discussed. [ABSTRACT FROM AUTHOR]

Let G be a finite group, H a subgroup of G and p a prime number. We say that H is weakly sp-permutable in G if G has a subnormal subgroup K such that G = HK, H s G ≤ K and | H ∩ K : H s G | is a p ′ -number, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of a group G under the assumption that certain subgroups of G are weakly sp-permutable in G. Some recent results are extended and generalized. Communicated by Alexander Olshanskii [ABSTRACT FROM AUTHOR]

In this paper, for any prime p, we propose the notion of a p-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique principal submodule to the case of the regular modules of modular adjacency algebras of association schemes. We characterize the p-transitive quasi-thin association schemes and the p-transitive association schemes with thin thin residue by their structure theory properties. We also get some results with independent interests. [ABSTRACT FROM AUTHOR]

The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups PSL (2 , p) , where p is a prime larger than 5, first partial results concerning the replacement property were published by Benjamin Nachman (2014). Second partial results were published by Hy P.G. Lam (2017). The main goal of this paper is to provide a complete answer for PSL (2 , p). [ABSTRACT FROM AUTHOR]

We previously have developed two methods for constructing codes and designs from finite groups (mostly simple finite groups). In a recent paper we introduced a new method (Method 3) for constructing codes and designs from fixed points of elements of finite transitive groups. This new method together with background material and results required from finite groups, permutation groups and representation theory were discussed fully in that paper. The main aim of this paper is to apply the method to the first Janko sporadic simple group J 1 , for the first 3 conjugacy classes of maximal subgroups and its 2A, 3A, 5A and 7A conjugacy classes of elements. [ABSTRACT FROM AUTHOR]