Zeros of characters and the Frattini subgroup (Q923125)

The author proves the following theorem: Let G be a finite group with \(1<Z(G)<G\), Z(G) the center of G, and assume G has an irreducible non- linear character \(\chi\) such that A(\(\chi\)) contains fewer than \(| Z(G)|\) conjugate classes of elements where \(A(\chi)=\{g\in G|\chi (g)=0\}\). Then the Frattini subgroup \(\Phi\) (G)\(\neq 1\).