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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]