Finite groups and subgroup-permutability
For the purposes of this paper, a finite group \(G\) is defined to be irreducible if the permutability graph on non-normal subgroups is connected, and reducible otherwise. The maximum diameter of the components of this graph is denoted \(\Delta(G)\). The four main results of the paper are: A: If \(G\) is soluble, then \(\Delta(G)\leq 4\). B: If \(G\) is non-abelian simple, then it is irreducible and \(\Delta(G)\leq 16\). C: If \(G\) is reducible, then it is soluble. D: If \(G\) is non-soluble, then \(\Delta(G)\leq\max\{\Delta(S)\}+4\), as \(S\) ranges over all non-abelian simple groups. The proof of B, C and D uses the classification theorem for finite simple groups.
permutability graph on non-normal subgroups
scientific article; zbMATH DE number 878432
scientific article