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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 it is proved that a finite group $G$ is $p$ -nilpotent provided $p$ is the smallest prime number dividing the order of $G$ and every minimal subgroup of $P \cap G^{'}$ is weakly-supplemented in $N_{G} (P),$ where $P$ is a Sylow $p$ -subgroup of $G$ . As applications, some interesting results with weakly-supplemented minimal subgroups of $P \cap G^{'}$ are obtained.

The Join of Two Fitting Classes.

Elspeth Cusack (1979)

Mathematische Zeitschrift

The lattice of Fitting classes which are right extensible by soluble groups.

M.J. Iranzo, Julio P. Lafuente, F. Pérez-Monasor (2005)

Publicacions Matemàtiques

In this paper we study the set of Fitting classes which are right extensible by soluble groups ordered by the inclusion relation. The consideration of the associated lattices gives rise to new Fitting classes and it allows to obtain some injectivity criteria for general Fitting classes.

The lattice of very-well-placed subgroups

Martha Saboyá-Baquero (1996)

Rendiconti del Seminario Matematico della Università di Padova

The $p$ -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

For a finite group $G$ and a fixed Sylow $p$ -subgroup $P$ of $G$ , Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$ -nilpotent if every element of $P \cap G^{'}$ with order $p$ lies in the center of $N_{G} (P)$ and when $p = 2$ , either every element of $P \cap G^{'}$ with order $4$ lies in the center of $N_{G} (P)$ or $P$ is quaternion-free and $N_{G} (P)$ is $2$ -nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$ -nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p = 2$ , every element of $P$ with order $4$ is also...