Strongly connective degree sets II: perfect derived subgroups

Abstract.

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G)  =  {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set ${\user1{\mathcal{C}}}$ of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of ${\user1{\mathcal{C}}}$, and every vertex outside of ${\user1{\mathcal{C}}}$ is adjacent to some member of ${\user1{\mathcal{C}}}$. When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect.