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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]

In this note, we prove that for every integer d ≥ 2 which is not a prime power, there exists a finite solvable group G such that d ∣ | G | , π (G) = π (d) and G has no subgroup of order d. We also introduce the CLT-degree of a finite group and answer two questions about it. [ABSTRACT FROM AUTHOR]

Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G , then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian. [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 G be a finite group and N be a nontrivial normal subgroup of G. Let Irr (G | N) be the set of all irreducible characters whose kernel does not contain N as a subgroup and Irr 2 (G | N) be the set of all linear and even degree characters of Irr (G | N) . By acd 2 (G | N) we mean the average degree of irreducible characters of Irr 2 (G | N) . Let Irr 2 (G | N) ≠ ∅ . We prove that N is solvable if acd 2 (G | N) ≤ 5 / 2 . Also, we prove the solvability of G by assuming acd 2 (G | N) < 5 / 2 . Moreover, the bounds are sharp. [ABSTRACT FROM AUTHOR]

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper, we classify these rings and show that all finite rings of order p n for n < 5 and some of order p 6 are in this class of rings. [ABSTRACT FROM AUTHOR]

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]

We prove that in a divisible -rigid group the algebraic closure of each set generating a subgroup of solvability length coincides with the elementary closure of this set. [ABSTRACT FROM AUTHOR]

We find all finite almost simple groups without elements of order and describe finite solvable groups without elements of order for an odd prime . [ABSTRACT FROM AUTHOR]

Let G be a finite group and C(G) denote the set of all non-normal non-cyclic subgroups of G. In this paper, the function dc(G) = 1 |G| P H∈C(G) |H| is introduced. In fact, we prove that, if dc(G) = 10 3, then either G ≤= A5, or G is solvable. We also find some examples of finite groups G with dc(G) ≤ 10 3. [ABSTRACT FROM AUTHOR]

A skew morphism of a finite group A is a permutation φ of A fixing the identity element and for which there is an integer-valued function π on A such that φ (a b) = φ (a) φ π (a) (b) for all a , b ∈ A . A skew morphism φ of A is smooth if the associated power function π is constant on the orbits of φ , that is, π (φ (a)) ≡ π (a) (mod | φ |) for all a ∈ A . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if n = 2 e n 1 , where 0 ≤ e ≤ 4 and n 1 is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given. [ABSTRACT FROM AUTHOR]

Let G be the semidirect product N ⋊ R , where N is a stratified Lie group and R acts on N via automorphic dilations. Homogeneous left-invariant sub-Laplacians on N and R can be lifted to G, and their sum Δ is a left-invariant sub-Laplacian on G. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin–Hörmander type was proved for Δ , showing that an operator of the form F (Δ) is of weak type (1, 1) and bounded on L p (G) for all p ∈ (1 , ∞) provided F satisfies a scale-invariant smoothness condition of order s > (Q + 1) / 2 , where Q is the homogeneous dimension of N. Here we show that, if N is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold s > (d + 1) / 2 , where d is the topological dimension of N. The proof is based on lifting to G weighted Plancherel estimates on N and exploits a relation between the functional calculi for Δ and analogous operators on semidirect extensions of Bessel–Kingman hypergroups. [ABSTRACT FROM AUTHOR]

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]

Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23.] showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism,$A_5$ is determined by its average order.

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 HG, where HG is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called a PHN-group if its all minimal subgroups and cyclic subgroup of order 4 are Hall normally embedded in G. In this paper, we give the classification of minimal non-PHN-groups. Furthermore, we investigate the structure of finite group all of whose maximal subgroups of even order are PHN-groups. [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]

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]

This article, titled "Reidemeister Classes, Wreath Products, and Solvability," discusses the concept of Reidemeister classes, which are equivalence classes of automorphisms of a group. The article explores three main directions of research related to Reidemeister classes: proving or disproving a conjecture about the relationship between Reidemeister numbers and irreducible unitary representations, searching for groups with infinite Reidemeister numbers, and studying the rationality and other properties of the Reidemeister zeta function. The article presents several theorems and lemmas related to these topics and provides proofs for some of them. The author concludes by discussing the solvability of certain groups and acknowledging funding support. [Extracted from the article]

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]

We show that if G is a finite group with fewer than 15 involutions, then Sol (G) ≠ 1 . We then show that if G contains fewer than 15 involutions such that O 2 (G) = 1 , then G is a solvable group of 2-rank 1. Furthermore, we show that if G contains n ∈ { 3 , 5 , 7 } involutions and O 2 (G) = 1 , then G is a supersolvable group, but it is not necessarily true for n ∈ { 9 , 11 , 13 } . Finally, we show that a non-solvable group generated by involutions has at least 15 involutions. [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]

Introduction Let G be a finite group and let Irr(G) be the set of irreducible characters of G. We say that an element g in G is a vanishing element if there exists some XE Irr(G) such that x(g) = 0. In this paper, we investigate the influence of the number of the columns containing some zeros in character table of G on the algebraic structure of G. Material and Methods Let Van(G) be the set of all vanishing elements, in other words, Van(G) = {g E G❘ 3χ € Irr(G) X(g) = 0 }. We can easily check that Van(G) is the union of some conjugacy classes. Burnside show that Van(G) = Ø if and only if G is an abelian group. We know that finite groups whose set of vanishing elements is the union of at most three conjugacy classes are solvable. Using this result, we provide a relatively short proof for the main theorem. Results and discussion In this paper, we classify finite groups whose set of vanishing elements is the union of exactly three conjugacy classes. Conclusion If the set of vanishing elements of a finite group G is the union of exactly three conjugacy classes then one of the following,situations:occurs Gis isomorphic to D8, Q8, or S4. G is a Frobenius group with abelian kernel of odd order and cyclic complement of order 4. GFxZ3, in which F is a Frobenius group with abelian kernel of odd order and complement of order 2. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N is non-abelian and N contains all but four conjugacy classes of G. [ABSTRACT FROM AUTHOR]

We denote the integer Πg∊G o(g) by ψ ′ (G) where o(g) denotes the order of g ∊ G and G is a finite group. In [14], it was proved that some finite simple group can be uniquely determined by its product of element orders. In this paper, we characterize Alt(5)×Zp, where p ∊ {17, 23}, by their product of element orders. [ABSTRACT FROM AUTHOR]

We prove that if H is a subgroup of a finite solvable group G such that the number of isotopic classes of right transversals of H in G is 1, then H is normal in G. [ABSTRACT FROM AUTHOR]

If G is a finite group and x ∈ G , we say that x lies in a small class if | x G | is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra. [ABSTRACT FROM AUTHOR]

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.

It is shown that, for any uncountable solvable group (G , ⋅) and any nonzero σ-finite left G-quasi-invariant measure μ on G, there exist two G-absolutely negligible subsets A and B of G such that the product A ⋅ B is not measurable with respect to μ. [ABSTRACT FROM AUTHOR]

Let be a finite group. For every element , the set is called the conjugacy class of in and is denoted by . The conjugacy class size of in is denoted by and is equal to . An element of is said to be primary or biprimary if the order of is divisible by exactly one or two distinct primes. For a positive integer and a prime , if is an integer such that divides and does not divide , then is called the -part of . Let be a prime divisor of such that . We prove that is solvable and -nilpotent if the conjugacy sizes of all noncentral primary and biprimary elements in have the same -part. On the other hand, suppose that is a normal subgroup of . We write . Suppose that , where . Denote by a subgroup of . We prove that if and is a normal subset of for every with , then is a nilpotent group with nilpotency class at most 2. [ABSTRACT FROM AUTHOR]

We provide a new and simple proof to show that a finite group in which every non-nilpotent maximal subgroup is normal is solvable. [ABSTRACT FROM AUTHOR]

We present a family of counterexamples to a question proposed recently by Moretó concerning the character codegrees and the element orders of a finite solvable group. [ABSTRACT FROM AUTHOR]

The classification of p-groups is a difficult task. It is also important because these groups have certain axioms and properties of which one can have further investigation for the other finite groups. In fact, there is no complete classification for p-groups, and the only complete classification has been done for certain cases of prime numbers or for certain nilpotency classes. In this article, we give a complete description of p-groups of order p4, where p is an odd prime. In addition, we discuss many other algebraic properties. Significantly, computer calculations were of an extensive use in this work. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G : ker φ|p′ = φ(1)p′ for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p′ is the p′-part of φ(1). [ABSTRACT FROM AUTHOR]

Every finite solvable group G has a normal series with nilpotent factors. The smallest possible number of factors in such a series is called the Fitting height h(G). In the present paper, we derive an upper bound for h(G) in terms of the exponent of G. Our bound constitutes a considerable improvement of an earlier bound obtained in Shalev (Proc Am Math Soc 126(12):3495–3499, 1998). [ABSTRACT FROM AUTHOR]

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

Let p be a prime and let G be a finite group which is generated by the set G p of its p -elements. We show that if G is solvable and not a p -group, then the minimal number σ p (G) of proper subgroups of G whose union contains G p is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p -solvable groups G , we always have σ p (G) ≥ p + 1. We study the case of alternating and symmetric groups G in detail. [ABSTRACT FROM AUTHOR]

Let G be a finite solvable group and let p be a prime divisor of $|G|$. We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character. [ABSTRACT FROM AUTHOR]

A group G is called to be EMN-group (rep. ESMN-group) if all maximal subgroup (rep. second maximal subgroup) of G of even order are nilpotent. In this paper, we mainly investigate the structure of EMN-groups and ESMN-groups. [ABSTRACT FROM AUTHOR]

Let p be a prime number. A positive integer m is said to be a p ′ -number provided p ∤ m . Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, H sG ≤ K and | H ∩ K : H sG | is a p ′ -number, where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s 2 -permutable in G. [ABSTRACT FROM AUTHOR]

A locally compact group G is called compactly covered (respectively compactly ruled) if it is an union of compact subgroups (resp. if it is a directed union of compact subgroups). The class [ CR ] of all compactly ruled groups is a proper subclass of the class [ CC ] of all compactly covered groups. In this paper we shall study the following question: For which classes X of locally compact groups do the following holds: X ∩ [ CC ] ⊆ [ CR ]. [ABSTRACT FROM AUTHOR]

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k, then the number of prime divisors of | G | {|G|} is bounded in terms of k. We prove this conjecture for solvable groups. [ABSTRACT FROM AUTHOR]

A subgroup of a finite group is said to be -subnormal if it is subnormal in , where is the generalized Fitting subgroup of . In the paper, the structure of groups factorizable by two -subnormal subgroups, one of which is nilpotent, is studied. In particular, if the other factor is metanilpotent, then the group is solvable. Moreover, if the commutator subgroup of the second factor is nilpotent, then the nilpotent length of the group is at most 3. The supersolvability of the product of a nilpotent subgroup by supersolvable subgroups and (all three are -subnormal) is established. [ABSTRACT FROM AUTHOR]

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V , and G is not metacyclic. Then G always has a regular orbit on V except for a few "small" cases. [ABSTRACT FROM AUTHOR]

We describe the finitely generated groups that are universally equivalent to the solvable Baumslag–Soliter group with parameters , . [ABSTRACT FROM AUTHOR]

Let G be a finite group and cd-(G) be the set of nonlinear irreducible character degrees of G. Suppose that -(G) denotes the set of primes dividing some element of cd-(G). The bipartite divisor graph for the set of character degrees which is denoted by B(G), is a bipartite graph whose vertices are the disjoint union of -(G) and cd-(G), and a vertex p 2 -(G) is connected to a vertex a 2 cd-(G) if and only if p|a. In this paper, we investigate the structure of a group G whose graph B(G) has five vertices. Especially we show that all these groups are solvable. [ABSTRACT FROM AUTHOR]

An automorphism $\alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_n\alpha]=1$ for all $g\in G$‎. ‎In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups‎. ‎We also show that‎, ‎assuming the truth of a conjecture about the representation theory of solvable groups raised by P‎. ‎Neumann‎, ‎it is possible to produce‎, ‎for a suitable prime $p$‎, ‎an example of a f.g‎. ‎solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group‎. ‎Conversely we show that‎, ‎if there exists a f.g‎. ‎solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms‎, ‎then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent‎.