Solvable groups whose character degree graphs generalize squares

Abstract

Abstract Let G be a solvable group, and let Δ ⁢ ( G ) {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that Δ ⁢ ( G ) {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and Δ ⁢ ( G ) {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.