Indices of elements and normal structure of finite groups. (Q1763739)

For a finite group \(G\) and for \(x\in G\), the authors use \(i_G(x)\) to denote the size of the conjugacy class of \(x\) in \(G\). They prove several powerful and elegant theorems, of which only two are stated in detail here. Theorem 1: Let \(P\in\text{Syl}_p(G)\) such that for all \(x\in P\) \(i_G(x)=p^k\), where \(k=0,1\) for \(p\) odd and \(k=0,1,2\) for \(p=2\). Then \(G\) is \(p\)-nilpotent. Theorem 2. If \(P\in\text{Syl}_p(G)\) is nonabelian and if \(i_G(x)\) is a \(p\)-power for all \(x\in P\setminus Z(P)\), then \(P\) is a direct factor of \(G\) and conversely. Theorem 3 characterizes the finite groups all whose elements of prime power order have conjugacy class sizes also prime powers -- this is a result of \textit{R. Baer} [Trans. Am. Math. Soc. 75, 20--47 (1953; Zbl 0051.25702)] obtained here with a short proof. The reader finds many more results (some on characters, some on groups all whose nonnilpotent subgroups \(H\), with \(|\pi(H)|=2\) are subnormal, etc.) of independent interest scattered throughout this well written and easy to read paper, which ends with seven open questions.