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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 theorem of Frobenius.

Bakić, Radoš (1997)

Publications de l'Institut Mathématique. Nouvelle Série

On Alperin's fusion theorem.

Stellmacher, Bernd (1994)

Beiträge zur Algebra und Geometrie

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 complete nilpotent groups.

А.Л. Шмелькин (1967)

Algebra i Logika

On cover-avoiding subgroups of Sylow subgroups of finite groups

Yangming Li (2010)

Rendiconti del Seminario Matematico della Università di Padova

On E-S-supplemented subgroups of finite groups

Changwen Li, Xuemei Zhang, Xiaolan Yi (2013)

Colloquium Mathematicae

The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini G-chief factor of a normal subgroup E of a finite group G is cyclic. As applications, some recent known results are generalized and unified.

On finite groups with independent subgroups.

Zhurtov, A.H., Tsirkhov, A.A. (2010)

Vladikavkazskiĭ Matematicheskiĭ Zhurnal

On finite groups with some subgroups of prime indices.

Monakhov, V.S., Tyutyanov, V.N. (2007)

Sibirskij Matematicheskij Zhurnal

On Finite Groups with Sylow 2-Subgroups of Type An, n = 8, 9, 10, 11.

Daniel Gorenstein, Koichiro Harada (1970)

Mathematische Zeitschrift

On Groups of Squarefree Order.

M.Ram Murty, V.Kumar Murty (1984)

Mathematische Annalen

On Hall subgroups of a finite group

Wenbin Guo, Alexander Skiba (2013)

Open Mathematics

New criteria of existence and conjugacy of Hall subgroups of finite groups are given.

On lattice properties of S-permutably embedded subgroups of finite soluble groups

L. M. Ezquerro, M. Gómez-Fernández, X. Soler-Escrivà (2005)

Bollettino dell'Unione Matematica Italiana

In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and $H \in {Hall}_{π} (G)$ such that $H \cap V \in {Hall}_{π} (V)$ and $1 \neq H \cap U \in {Hall}_{π} (U)$ . Suppose also $H \cap U$ is a Hall π-sub-group of some S-permutable subgroup of G. Then $H \cap U \cap V \in {Hall}_{π} (U \cap V)$ and $〈 H \cap U, H \cap V 〉 \in {Hall}_{π} (〈 U \cap V 〉)$ . Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.

Currently displaying 1 – 20 of 52

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Displaying 1 – 20 of 52

Odd Automorphisms of Sylow 2-subgroups of sporadic simple groups.

Odd Order Hall Subgroups of GL(n, q) and Sp(2n, q).

Odd order Hall subgroups of the classical linear groups.

On a class of groups with Lagrangian factor groups.

On a generalization of a theorem of Burnside