Zur Vertauschbarkeit und Subnormalität von Untergruppen. (On the permutability and subnormality of subgroups) (Q1120673)

The author considers the following embedding properties of a subgroup X in a finite group G: X is permutable with all, (M) maximal subgroups, \((M^*)\) maximal subgroups of subnormal subgroups containing X, (N) selfnormalizing subgroups, of G respectively. He is able to show that N- embedded groups X are also \(M^*\)-embedded and subnormal (Satz 4.5); its maximal perfect subgroup is normal in G (Satz 4.7). On the other hand for every non-nilpotent group X there is a non-subnormal M-embedding in some G such that \(| G:X|\) is a power of a suitable prime dividing \(| X|\) (Satz 3.3) while for nilpotent groups all M-embeddings are subnormal embeddings (Folgerung 3.4). The situation is explained further by examples.