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A proper subgroup H of a finite group G is called a second maximal subgroup of G if H is a maximal subgroup of every maximal subgroup M in G with H ≤ M. In this paper, we investigate the structure of the finite group G in which CG(x) is a second maximal subgroup for every non-central element x in G, and prove that either G is solvable or G/Z(G)≅PSL(2,rn), where r = 2 with n a prime or r = 3 with n an odd prime. [ABSTRACT FROM AUTHOR]

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]

A subgroup 퐻 of a group 퐺 is pronormal if, for each 푔 퐺 , the subgroups 퐻 and superscript 퐻 푔 are conjugate in 퐻 superscript 퐻 푔 . Most of finite simple groups possess the following property ∗ : each subgroup of odd index is pronormal in the group. The conjecture that all finite simple groups possess the property ∗ was established in 2012 in a paper by E.P. Vdovin and the third author based on the analysis of the proof that Hall subgroups are pronormal in finite simple groups. However, the conjecture was disproved in 2016 by A.S. Kondrat'ev together with the second and third authors. In a series of papers by Kondrat'ev and the authors published from 2015 to 2020, the finite simple groups with the property ∗ except finite simple linear and unitary groups with some constraints on natural arithmetic parameters were classified. In this paper, we construct series of examples of nonpronormal subgroups of odd index in finite simple linear and unitary groups over a field of odd characteristic, thereby making a step towards completing the classification of finite simple groups with the property ∗ . [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 paper, we give a solvability criterion for a finite group G depending on the minimum of the ratio χ(1)2/cod(χ), when χ varies among the irreducible characters 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) . Let b c (G) be the largest codegree of G. In this paper, we study how the structure of a group G is bounded by its largest codegree b c (G) . Firstly, we give a criterion for solvability: If b c (G) < 20 , then G is solvable. Secondly, we consider the nonsolvable groups G with a small b c (G) and prove that, if b c (G) < 56 , then G is isomorphic to A 5 , S 5 or A 5 × A where A is an elementary abelian 2-group. [ABSTRACT FROM AUTHOR]

Given a group G, we define the power graph P(G) as follows: the vertices are the elements of G and two vertices x and y are joined by an edge if ⟨x⟩ ⊆ ⟨y⟩ or ⟨y⟩ ⊆ ⟨x⟩. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. We consider κ(G), the number of spanning trees of the power graph associated with a finite group G. In this paper, for a finite group G, first we represent some properties of P(G), then we are going to find some divisors of κ(G), and finally we prove that the simple group A6 ∼= L2(9) is uniquely determined by tree-number of its power graph among all finite simple groups.

Let G be a finite group, F be one of the fields Q , R or C , and N be a non-trivial normal subgroup of G. Let acd F ∗ (G) and acd F , even (G | N) be the average degree of all non-linear F -valued irreducible characters of G and of even degree F -valued irreducible characters of G whose kernels do not contain N, respectively. We assume the average of an empty set is zero for more convenience. In this paper we prove that if acd Q ∗ (G) < 9 / 2 or 0 < acd Q , even (G | N) < 4 , then G is solvable. Moreover, setting F ∈ { R , C } , we obtain the solvability of G by assuming acd F ∗ (G) < 29 / 8 or 0 < acd F , even (G | N) < 7 / 2 , and we conclude the solvability of N when 0 < acd F , even (G | N) < 18 / 5 . Replacing N by G in acd F , even (G | N) gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp. [ABSTRACT FROM AUTHOR]

For a finite group G, the superpower graph S(G) of G is an undirected simple graph with vertex set G and two vertices are adjacent in S(G) if and only if the order of one divides the order of the other in G. The aim of this paper is to provide tight bounds for the vertex connectivity, discuss Hamiltonian-like properties of superpower graph of finite non-Abelian groups having an element of exponent order.We also give some general results about superpower graphs and their relation to other graphs such as the Gruenberg–Kegel graph. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We denote by ψ ′ (G) the integer ∏ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. It is verified that some finite groups are uniquely determined by their orders and their product of element orders. In this paper, we prove that, if p , q , r , t are different prime numbers, then ℤ p 3 , ℤ p q , ℤ p 2 q and ℤ p q r t are characterized by ψ ′ (G) and if ψ ′ (G) = ψ ′ (ℤ p q r) , then G ≅ A 5 or ℤ p q r . [ABSTRACT FROM AUTHOR]

Given a group G, we define the power graph P(G) as follows: the vertices are the elements of G and two vertices x and y are joined by an edge if ⟨x⟩ ⊆ ⟨y⟩ or ⟨y⟩ ⊆ ⟨x⟩. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. We consider κ(G), the number of spanning trees of the power graph associated with a finite group G. In this paper, for a finite group G, first we represent some properties of P(G), then we are going to find some divisors of κ(G), and finally we prove that the simple group A6= L2(9) is uniquely determined by tree-number of its power graph among all finite simple groups. [ABSTRACT FROM AUTHOR]

Let $ G $ be a finite group, $ \operatorname{cd}(G) $ the set of all irreducible character degrees of $ G $, and $ \rho(G) $ the set of all prime divisors of integers in $ \operatorname{cd}(G) $. For a prime $ p $ and a positive integer $ n $, let $ n_p $ denote the $ p $-part of $ n $. The degree prime-power graph of $ G $ is a graph whose vertex set is $ V(G) = \left\{p^{e_p(G)} \mid p \in \rho(G)\right\} $, where $ p^{e_p(G)} = \max \left\{n_p \mid n \in \operatorname{cd}(G)\right\} $, and there is an edge between distinct numbers $ x, y \in V(G) $ if $ x y $ divides some integer in $ \operatorname{cd}(G) $. The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group $ PSp_4(p) $ can be uniquely identified by its order and degree prime-power graph.

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 G be a finite group, cd(G) the set of all irreducible character degrees of G, and p(G) the set of all prime divisors of integers in cd(G). For a prime p and a positive integer n, let np denote the p-part of n. The degree prime-power graph of G is a graph whose vertex set is V(G) = {...| p ∈ p(G)}, where ... = max {np | n ∈ cd(G)}, and there is an edge between distinct numbers x, y ∈ V(G) if xy divides some integer in cd(G). The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group PS p4(p) can be uniquely identified by its order and degree prime-power graph. [ABSTRACT FROM AUTHOR]

Let G p be the set of p -elements of a finite group G. Do we need all the Sylow p -subgroups of G to cover G p ? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect. [ABSTRACT FROM AUTHOR]

This paper mainly presents series of research work on Sylow's theorem and examines its formal proof using concrete examples for verification and identifying some of its consequences. Furthe1more, some of its applications on a cyclic group order, small (but not simple) groups and infinite groups. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p > 0. We prove that if all irreducible representations over K of a finite group G have degree at most n , then G has a characteristic subgroup N of index bounded above in terms of n such that its derived subgroup N ′ is a p -group. Our proof of this result relies on the celebrated Larsen-Pink theorem. We also show that every finite group G has a solvable p -nilpotent subgroup of index bounded above in terms of the largest p ′ -degree of the complex irreducible characters of G. [ABSTRACT FROM AUTHOR]

In the present article, we prove that every countable infinite group is embeddable into a countable infinite simple group such that every equation of the form is solvable in , where is a nontrivial reduced group word in variables and . [ABSTRACT FROM AUTHOR]

Let G be a finite group. We say that G has the Basis Property if every subgroup H of G has a minimal generating set (basis), and any two bases of H have the same cardinality. A group G is called minimal not satisfying the Basis Property if it does not satisfy the Basis Property, but all its proper subgroups satisfy the Basis Property. We prove that the following groups P SL(2, 5) ∼= A5, P SL(2, 8), are minimal groups non satisfying the Basis Property, but the groups P SL(2, 9), P SL(2, 17) and P SL(3, 4) are not minimal and not satisfying the Basis Property. [ABSTRACT FROM AUTHOR]

For a 1-$(v,k,\lambda)$ design $\mathcal{D}$ containing a point $x$, we study the set $I_x$, the intersection of all blocks of $\mathcal{D}$ containing $x$. We use the set $I_x$ together with the Key-Moori Method 2 to construct reduced designs invariant under some families of finite simple groups. We also show that there is a connection between reduced designs constructed by Method 2 and the new Moori Method 3.

In this paper, the structures of non-solvable groups whose all proper subgroups have at most seven character values are identified.

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. In this paper, we classify all connected cubic symmetric graphs of order 52p2 for each prime p.

Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.

The theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota–Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota–Baxter operator. Thus, a group endowed with a Rota–Baxter operator gives rise to a set-theoretical solution to the Yang–Baxter equation. We provide some general constructions of Rota–Baxter operators on a group. Given a map on a group, we study its extensions to a Rota–Baxter operator. We state the connection between Rota–Baxter operators on a group and Rota–Baxter operators on an associated Lie ring. We describe Rota–Baxter operators on sporadic simple groups. [ABSTRACT FROM AUTHOR]

A permutation group G on a set Ω is called orbit closed if every permutation of Ω preserving the orbits of G in its action on the power set P (Ω) belongs to G. It is called a relation group if there exists a family R ⊆ P (Ω) such that G is the group of all permutations preserving R. We prove that if a finite orbit closed permutation group G is simple, or is a subgroup of a simple group, then it is a relation group. This result justifies our general conjecture that with a few exceptions every finite orbit closed group is a relation group. To obtain the result, we prove that most of the finite simple permutation groups are relation groups. We also obtain a complete description of those finite simple permutation groups that have regular sets, and prove that (with one exception) if a finite simple permutation group G is a relation group, then every subgroup of G is a relation group. [ABSTRACT FROM AUTHOR]

In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of 2-power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property.

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]). Let Γ be a connected symmetric graph of prime valency p ≥ 5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph Cay (M , S) such that M is a 2-group admitting a fixed-point-free automorphism of order p with S as an orbit of involutions, or Γ is a normal N -cover of a T -arc-transitive graph of valency p admitting a quasi-semiregular automorphism, where T is a non-abelian simple group and N is a nilpotent group. Further, for p = 5 a complete classification of graphs Γ such that either Aut (Γ) has a solvable arc-transitive subgroup or Γ is T -arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given. [ABSTRACT FROM AUTHOR]

Let G be a group, define an equivalence relation ~ as below: ∀ g, h ∈ G, g ∼ h ⇐⇒|g| = |h| the set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denoted by α(G). And G is said a αn-group if |α(G)| = n. Let π(G) be the set of prime divisors of the order of G. A simple group of G is called a simple Kn-group if |π(G)| = n. We give a new characterization of simple K3-groups using same-order type. Indeed we prove that a nonabelian simple group G has same-order type {r, m, n, k, l} if and only if G ≅ PSL(2,q), with q = 7, 8 or 9. This result generalizes the main results in (4), (6) and (8). Moreover based on the main result in (8) we have the natural question: Let S be a nonabelian simple αn-group and G a αn-group such that |S| = |G|. Then S ≅ G. In this paper with a counterexample we give a negative answer to this question. [ABSTRACT FROM AUTHOR]

In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of 2 -power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property. [ABSTRACT FROM AUTHOR]

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A regular graph is semisymmetric if its automorphism group acts transitively on the edge set but not on the vertex set. In this article, we give a complete list of connected semisymmetric graphs of square‐free order and valency 5. The list consists of a single graph, the incidence graph of a generalized hexagon of order (4,4) $(4,4)$, and an infinite family arising from some groups with cyclic Fitting subgroup. [ABSTRACT FROM AUTHOR]

For a positive integer n and a prime p, let np denote the p-part of n. Let G be a group, cd (G) the set of all irreducible character degrees of G, ρ (G) the set of all prime divisors of integers in cd (G) , V (G) = { p e p (G) ∣ p ∈ ρ (G) } , where p e p (G) = max { χ (1) p ∣ χ ∈ Irr (G) }. The authors proved that G ≅ L 2 (p 2) if and only if | G | = | L 2 (p 2) | and V (G) = V (L 2 (p 2)). In this paper, the authors continue this topic and prove that G ≅ L 2 (p 3) if and only if | G | = | L 2 (p 3) | and V (G) = V (L 2 (p 3)). [ABSTRACT FROM AUTHOR]

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 M 11 , M 12 , M 22 , M 23 , and PSL (3 , 3) determines the group up to isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the finite group G is solvable if cod (χ) < χ α (1) for any nonlinear irreducible character χ of G where α ≈ 1.8876 . [ABSTRACT FROM AUTHOR]

We study the structure of nonsolvable finite groups whose every nonsolvable maximal subgroup is -simple. [ABSTRACT FROM AUTHOR]

In this paper we present a new sufficient condition for a solubility criterion in terms of centralizers of elements. This result is a corrigendum of one of Zarrin's results. Furthermore, we extend some of K. Khoramshahi and M. Zarrin's results in the primitive case. [ABSTRACT FROM AUTHOR]

Let G be a group. The prime index graph of G, denoted by Π(G), is an undirected graph whose vertices are all subgroups of G and two distinct comparable subgroups H and K are adjacent if and only if [H: K] or [K: H] is prime. In this paper among other results, it is shown that the prime index graph of a finite simple group G is connected if and only if G is isomorphic to A5, PSL2(11), PSL3(2), PSL3(3) or PSL2(22n), where n ≤ 4. [ABSTRACT FROM AUTHOR]

We characterize seven valent symmetric graphs of order 2pqn2p{q}^{n} with p

In this paper, the commuting graphs associated with finite groups are discussed. As the main theme, the explicit formulas for the number of spanning trees of commuting graphs associated with some specific groups, such as the Suzuki simple groups, the projective special linear groups, and some certain AC-groups, are obtained. [ABSTRACT FROM AUTHOR]

This paper is a new important step towards the complete classification of the finite simple groups which are (2 , 3) -generated. In fact, we prove that the symplectic groups Sp 2 n (q) are (2 , 3) -generated for all n ≥ 4. Because of the existing literature, this result implies that the groups PSp 2 n (q) are (2 , 3) -generated for all n ≥ 2 , with the exception of PSp 4 (2 f) and PSp 4 (3 f). [ABSTRACT FROM AUTHOR]

In this paper, we will continue to discuss the topic on NS-supplemented subgroups proposed by Monakhov and Trofimuk. In detail, we obtained the classification of nonabelian pd-composition factors by using the NS-supplemented Sylow p-subgroup (or NS-supplemented sylowizer for every maximal subgroup of a Sylow p-subgroup) and we also investigated p-supersolvable hypercentre Z p U (G) of a group G. [ABSTRACT FROM AUTHOR]

In 2020, the first author and Pan proved that every edge-primitive graph of valency 6 is 2-arc-transitive, and except the complete bipartite graph K 6 , 6 , the automorphism group is almost simple, and they determined such graphs having a solvable edge stabilizer. The nonsolvable edge stabilizer case is settled in this work, which leads to a complete classification of edge-primitive graphs of valency 6. [ABSTRACT FROM AUTHOR]

Let a group A act on the group G coprimely. Suppose that the order of the fixed point subgroup C G (A) is not divisible by an arbitrary but fixed prime p. In the present paper we determine bounds for the p -length of the group G in terms of the order of A , and investigate how some A -invariant p -subgroups are embedded in G under various additional assumptions. [ABSTRACT FROM AUTHOR]

The spectrum of a finite group is the set of its element orders. H is said to be a finite cover of G if G is a homomorphic image of H and H is finite. The main aim of this article is to characterize the finite simple groups with no elements of order 10 by its spectrum among covers. At the same time, above simple groups are completely classified. At last, some results on the recognition by spectrum of above groups are also achieved. [ABSTRACT FROM AUTHOR]