On a Fitting length conjecture without the coprimeness condition

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ≅ A 5.

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we generalize a result of J. G. Thompson and characterize finite solvable groups as those finite groups which do not contain nontrivial elements xi , i = 1,2,3, with x 1 x 2 x 3 = 1 and xi a pi -element for distinct primes pi . We note some connections with coverings of curves.

Let A be a finite group and let n be the number of primes, including multiplicities, which divide the order of A. Suppose further that G is a finite solvable group which admits A as a fixed-point-free operator group and such that (i A I, G|I ) = 1. Under these assumptions it has been conjectured that l(G), the nilpotent length of G, is at most n, while lp(G), the p-length of G, is at most [(n+1)/2]. These inequalities have been verified and shown to be best-possible in certain special cases ( [3 ][5], [1]). It is the purpose of this note to show that these conjectured upper bounds are actually obtained provided A is solvable. Specifically we will prove

Finite Groups With Many Conjugate Elements

The main result in this note is the followingTheorem: Let G be a finite solvable group. There exists a permutation σ of the set G such that {g • σ(g); g∈G} = G if and only if the Sylow 2-subgroup of G is non-cyclic or trivial

Every finite abelian group is a subgroup of the additive group of a finite simple left brace

An answer to a question of J.G. Thompson on some generalized characters

Blocks of small defect

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \({\mathfrak {m}_{p}(G)}\) , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.

Regular and p-regular orbits of solvable linear groups

An automorphism $\\alpha$ of an integral group ring $\\mathbb{Z} G$, where $G$ is a finite group, is said to have a Zassenhaus factorization if it is the composition of an automorphism of $G$ (extended to a ring automorphism) and a central automorphism. In 1988, Roggenkamp and Scott constructed a group $G$ (of order $2880$) such that $\\mathbb{Z} G$ has a normalized (i.e., augmentation preserving) automorphism $\\alpha$ which has no Zassenhaus factorization. In this paper, short proofs of the following two results are given. (1) For a group $G$ of order $144$, there is a normalized automorphism $\\alpha$ of $\\mathbb{Z} G$ which has no Zassenhaus factorization. Moreover, $\\alpha$ can be chosen to have finite order. (2) There is a group $G$ of order $1200$, with abelian Sylow subgroups and Sylow tower, such that $\\mathbb{Z} G$ has a normalized automorphism which has no Zassenhaus factorization.

Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian [Formula: see text]-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively.

Let F be a free group of arbitrary rank and let H be a finitely generated subgroup of F. Given a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Our main result states that it is decidable whether H is pro-Su dense, where Su⊂S denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether H is pro-S dense?

It is proved that finite groups with 10p elements of the largest order, wherep ≥ 23 a prime, are solvable in this article. Consequently it follows that all finite group with 10p elements of largest order are solvable by some other results.