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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 $G$ , $| G / K : N_{G / K} (H K / K \cap L / K) |$ is a $π (H K / K \cap L / K)$ -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 number theoretic approach to Sylow r-subgroups of classical groups.

Mashhour I. AlAli, Christoph Hering, Anni Neumann (2005)

Revista Matemática Complutense

The purpose of this paper is to give a general and a simple approach to describe the Sylow r-subgroups of classical groups.

A $p$ -stable action of the automorphism group of a group

M. D. Pérez-Ramos (1988)

Rendiconti del Seminario Matematico della Università di Padova

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 Theorem of Sylow Type for Finite Groups.

B. Hartley (1971)

Mathematische Zeitschrift

A $Z J$ -theorem for $p^{*}, p$ -injectors in finite groups

M. J. Iranzo, A. Martínez-Pastor, F. Pérez-Monasor (1992)

Rendiconti del Seminario Matematico della Università di Padova

Active sums of groups.

P. Ribenboim (1981)

Journal für die reine und angewandte Mathematik

Addendum to : “Prime graph components of finite almost simple groups” [Rend. Sem. Mat. Univ. Padova 102 (1999), 1–22]

Maria Silvia Lucido (2002)

Rendiconti del Seminario Matematico della Università di Padova

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.

An elementary study of groups whose order is a product of two primes.

Ríder Moyano, Alfonso, Rubio Ruiz, Rafael María (2003)

Divulgaciones Matemáticas

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