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A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group...

On a class of groups with Lagrangian factor groups.

Tuccillo, Fernando (1994)

Portugaliae Mathematica

On a definition for formations

Marco Barlotti (1988)

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

By constructing appropriate faithful simple modules for the group GL(2,3), the author shows that certain "local" definitions for formations are not equivalent.

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

A theorem of Burnside asserts that a finite group $G$ is $p$ -nilpotent if for some prime $p$ a Sylow $p$ -subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $| G |$ , the order of $G$ . Let $P \in {Syl}_{p} (G)$ . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$ -subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P ≇ 〈 a, b | a^{p^{n - 1}} = 1, b^{2} = 1, b^{- 1} a b = a^{1 + p^{n - 2}} 〉$ , where $n \geq 3$ for $p > 2$ and $n \geq 4$ for $p = 2$ , then $G$ is $p$ -nilpotent or $p$ -closed.

On a problem of the theory of multiply local formations.

Guo, Wenbin (2004)

Sibirskij Matematicheskij Zhurnal

On a question of Chambers and Makan.

A. Ballester-Bolinches, M. Pérez-Ramos (1996)

Mathematische Zeitschrift

On a theorem of Frobenius.

Bakić, Radoš (1997)

Publications de l'Institut Mathématique. Nouvelle Série

On abelian inner mapping groups of finite loops

Markku Niemenmaa (2000)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider finite loops of specific order and we show that certain abelian groups are not isomorphic to inner mapping groups of these loops. By using our results we are able to construct a finite solvable group of order 120 which is not isomorphic to the multiplication group of a finite loop.

On complemented subgroups of finite groups

Long Miao (2006)

Czechoslovak Mathematical Journal

A subgroup $H$ of a group $G$ is said to be complemented in $G$ if there exists a subgroup $K$ of $G$ such that $G = H K$ and $H \cap K = 1$ . In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about $p$ -nilpotent groups.

On cover-avoiding subgroups of Sylow subgroups of finite groups

Yangming Li (2010)

Rendiconti del Seminario Matematico della Università di Padova

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