SM-vanishing conjugacy classes of finite groups (Q6655939)

An element \(g\) of a finite group \(G\) is a SM-vanishing element of \(G\), if there exists a strongly monolithic character \(\chi\) of \(G\) such that \(\chi(g)= 0\). The conjugacy class \(g^{G}\) of a SM-vanishing element \(g\) is called an SM-vanishing conjugacy class of \(G\).\N\NLet \(G\) be a finite group and let \(p\) be a fixed prime number. In this paper the authors prove the following results.\N\NTheorem A: If every SM-vanishing conjugacy class size of \(G\) is not divisible by \(p\), then \(G\) has a normal \(p\)-complement.\N\NTheorem B: If every SM-vanishing conjugacy class size of \(G\) is a \(p\)-power, then \(G\) has a normal Sylow \(p\)-subgroup.\N\NTheorem C: Assume that every SM-vanishing conjugacy class size of \(G\) is a prime power. Then \(G/F(G)\) is abelian and so \(G\) is solvable.\N\NTheorem D: Assume that if \(q \in \pi(G)\), then \(q\) does not divide \(p-1\). If every SM-vanishing conjugacy class size of \(G\) is not divisible by \(p^{2}\), then \(G\) is a solvable group.