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Let $G$ be a group and $ω (G)$ be the set of element orders of $G$ . Let $k \in ω (G)$ and $m_{k} (G)$ be the number of elements of order $k$ in $G$ . Let nse $(G) = {m_{k} (G) : k \in ω (G)}$ . Assume $r$ is a prime number and let $G$ be a group such that nse $(G) =$ nse $(S_{r})$ , where $S_{r}$ is the symmetric group of degree $r$ . In this paper we prove that $G ≅ S_{r}$ , if $r$ divides the order of $G$ and $r^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

A note on a class of factorized $p$ -groups

Enrico Jabara (2005)

Czechoslovak Mathematical Journal

In this note we study finite $p$ -groups $G = A B$ admitting a factorization by an Abelian subgroup $A$ and a subgroup $B$ . As a consequence of our results we prove that if $B$ contains an Abelian subgroup of index $p^{n - 1}$ then $G$ has derived length at most $2 n$ .

A note on groups with hamiltonian quotients

R. A. Bryce, John Cossey (1998)

Rendiconti del Seminario Matematico della Università di Padova

A note on hamiltonian 2-groups

R. A. Bryce, John Cossey (1991)

Rendiconti del Seminario Matematico della Università di Padova

A note on loops of square-free order

Emma Leppälä, Markku Niemenmaa (2013)

Commentationes Mathematicae Universitatis Carolinae

Let $Q$ be a loop such that $| Q |$ is square-free and the inner mapping group $I (Q)$ is nilpotent. We show that $Q$ is centrally nilpotent of class at most two.

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 remark on operating groups.

Wang, Yanming (1997)

International Journal of Mathematics and Mathematical Sciences

A result about cosets

John C. Lennox, James Wiegold (1995)

Rendiconti del Seminario Matematico della Università di Padova

Abelian quasinormal subgroups of groups

Stewart E. Stonehewer, Giovanni Zacher (2004)

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

Let $G$ be any group and let $A$ be an abelian quasinormal subgroup of $G$ . If $n$ is any positive integer, either odd or divisible by $4$ , then we prove that the subgroup $A^{n}$ is also quasinormal in $G$ .

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2-Groups which Contain Exactly Three Involutions.

A Cohomological Property of Regular p-Groups.

A Frobenius Theorem for Blocks.

A generalized Frattini subgroup of a finite group.

A new characterization of symmetric group by NSE

A note on a class of factorized $p$ -groups

A note on groups with hamiltonian quotients

A note on hamiltonian 2-groups

A note on loops of square-free order

A note on the minimal normal Fitting class

A remark on operating groups.

A result about cosets

Abelian quasinormal subgroups of groups

About noncommuting graphs.

Algebre di Lie graduate in caratteristica due

An algorithm for the word problem in $H N N$ extensions and the dependence of its complexity on the group representation

Analoga des Fundamentalsatzes der projektiven Geometrie in der Gruppentheorie. II

Analoga des Fundamentalsatzes der projektiven Geometrie in der Gruppentheorie - I

Another Homological Characterisation of Finite p-Nilpotent Groups.

Automorfismi involutori di p-gruppi finiti

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