Classification of finite groups according to their conjugacy class lengths (Q1755298)

A finite group \(G\) is said to satisfy condition \(P_n\) if for every prime \(p\), \(G\) has at most \(n-1\) conjugacy classes whose sizess are multiples of \(p\). Therefore, if \(G\) does not satisfy \(P_n\), that is, if there exist \(n\) conjugacy classes in \(G\) whose sizes are divisible by some fixed primer \(p\), then these classes constitute a subgraph (isomorphic to the complete graph \(K_n\)) of the conjugacy class graph of \(G\). These conditions have been addresed for several authors. In [J. Algebra 264, No. 2, 613--619 (2003; Zbl 1024.20021)], \textit{M. Fang} and \textit{P. Zhang} classified those finite groups whose conjugacy class graphs have no subgraph \(K_3\). In [Arch. Math. 85, No. 2, 101--107 (2005; Zbl 1104.20027)], \textit{A. Moretó} et al. studied the structure of all finite groups satisfying condition \(P_5\). From this classification, it is easy to obtain those groups whose conjugacy class graph does not contain a subgraph \(K_4\) or \(K_5\), what extends the result of Fang and Zhang [loc. cit.]. In the paper under review, all finite groups that satisfy condition \(P_6\) and that do not satisfy condition \(P_5\) are classified.