ON -QUASINORMAL SUBGROUPS OF FINITE GROUPS

Abstract

Let $G$ be a finite group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes $\mathbb{P}$ , that is, $\mathbb{P}=\bigcup _{i\in I}\unicode[STIX]{x1D70E}_{i}$ and $\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}=\emptyset$ for all $i\neq j$ . We say that $G$ is $\unicode[STIX]{x1D70E}$ - primary if $G$ is a $\unicode[STIX]{x1D70E}_{i}$ -group for some $i$ . A subgroup $A$ of $G$ is said to be: $\unicode[STIX]{x1D70E}$ - subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D70E}$ -primary for all $i=1,\ldots ,n$ ; modular in $G$ if the following conditions hold: (i) $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ for all $X\leq G,Z\leq G$ such that $X\leq Z$ and (ii) $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ for all $Y\leq G,Z\leq G$ such that $A\leq Z$ ; and $\unicode[STIX]{x1D70E}$ - quasinormal in $G$ if $A$ is modular and <jats: