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A note on the minimal normal Fitting class

Marco Barlotti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Un gruppo finito ciclico-per-nilpotente appartiene alla minima classe di Fitting normale se e solo se è nilpotente.

A note on the Π -property of some subgroups of finite groups

Zhengtian Qiu, Guiyun Chen, Jianjun Liu (2024)

Czechoslovak Mathematical Journal

Let H be a subgroup of a finite group G . We say that H satisfies the Π -property in G if for any chief factor L / K of G , | G / K : N G / K ( H K / K ∩ L / K ) | is a π ( H K / K ∩ L / K ) -number. We obtain some criteria for the p -supersolubility or p -nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the Π -property.

A note on weakly-supplemented subgroups and the solvability of finite groups

Xin Liang, Baiyan Xu (2022)

Czechoslovak Mathematical Journal

Suppose that G is a finite group and H is a subgroup of G . The subgroup H is said to be weakly-supplemented in G if there exists a proper subgroup K of G such that G = H K . In this note, by using the weakly-supplemented subgroups, we point out several mistakes in the proof of Theorem 1.2 of Q. Zhou (2019) and give a counterexample.

A note on weakly-supplemented subgroups of finite groups

Hong Pan (2018)

Czechoslovak Mathematical Journal

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = H K . In the paper, we extend one main result of Kong and Liu (2014).

A remark on a Theorem of J. G. Thompson

Bertram Huppert (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

An important theorem by J. G. Thompson says that a finite group G is p -nilpotent if the prime p divides all degrees (larger than 1) of irreducible characters of G . Unlike many other cases, this theorem does not allow a similar statement for conjugacy classes. For we construct solvable groups of arbitrary p -lenght, in which the lenght of any conjugacy class of non central elements is divisible by p .

A result about cosets

John C. Lennox, James Wiegold (1995)

Rendiconti del Seminario Matematico della Università di Padova

A solvability criterion for finite groups related to character degrees

Babak Miraali, Sajjad Mahmood Robati (2020)

Czechoslovak Mathematical Journal

Let m > 1 be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly m prime divisors. We show that such groups are solvable whenever m > 2 . Moreover, we prove that if G is a non-solvable group with this property, then m = 2 and G is an extension of A 7 or S 7 by a solvable group.

Algorithms for permutability in finite groups

Adolfo Ballester-Bolinches, Enric Cosme-Llópez, Ramón Esteban-Romero (2013)

Open Mathematics

In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.

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