Finite groups whose noncentral class sizes have the same \(p\)-part for some prime \(p\).
A well-known theorem of Thompson states that if the prime \(p\) divides the degree of each nonlinear irreducible complex character of the finite group \(G\), then \(G\) is \(p\)-nilpotent. While it is known that this result in general has no analogue for conjugacy class sizes in place of character degrees, the paper under review presents a partial analogue as follows. If the sizes of the noncentral conjugacy classes of \(G\) have the same non-trivial \(p\)-part, then \(G\) is \(p\)-nilpotent. In addition, the authors prove that such groups are solvable of Fitting height no more than 3 and provide a lot more structural information on these groups. Moreover, they prove that if \(m\) is a positive integer such that the noncentral conjugacy class sizes \(|G:C_G(g)|\) (where \(g\in G-Z(G)\)) are all of the form \(mn_g\) for some integer \(n_g\) satisfying \((m,n_g)=1\), then \(m\) necessarily is a prime power. The authors remark that their proofs do not rely on CFSG.
scientific article; zbMATH DE number 6127522