Journal Article

ON THE LATTICE OF -SUBNORMAL SUBGROUPS OF A FINITE GROUP

SKIBA, ALEXANDER N., GUO, WENBIN

2017

Overview

Let$\\unicode[STIX]{x1D70E}=\\{\\unicode[STIX]{x1D70E}_{i}\\mid i\\in I\\}$be a partition of the set of all primes$\\mathbb{P}$. Let$\\unicode[STIX]{x1D70E}_{0}\\in \\unicode[STIX]{x1D6F1}\\subseteq \\unicode[STIX]{x1D70E}$and let$\\mathfrak{I}$be a class of finite$\\unicode[STIX]{x1D70E}_{0}$-groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group$G$is$\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$- primary provided$G$is either an$\\mathfrak{I}$-group or a$\\unicode[STIX]{x1D70E}_{i}$-group for some$\\unicode[STIX]{x1D70E}_{i}\\in \\unicode[STIX]{x1D6F1}\\setminus \\{\\unicode[STIX]{x1D70E}_{0}\\}$and we say that a subgroup$A$of an arbitrary group$G^{\\ast }$is$\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$- subnormal in$G^{\\ast }$if there is a subgroup chain$A=A_{0}\\leq A_{1}\\leq \\cdots \\leq A_{t}=G^{\\ast }$such that either$A_{i-1}\\unlhd A_{i}$or$A_{i}/(A_{i-1})_{A_{i}}$is$\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$-primary for all$i=1,\\ldots ,t$. We prove that the set${\\mathcal{L}}_{\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}}(G)$of all$\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$-subnormal subgroups of$G$forms a sublattice of the lattice of all subgroups of$G$and we describe the conditions under which the lattice${\\mathcal{L}}_{\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}}(G)$is modular.

• Publisher: Cambridge University Press
• Subject: Subgroups