Groups with relatively small normalizers of primary subgroups. (Q1936275)

The paper under review presents a classification of those finite groups \(G\) in which, for every subgroup \(A\leq G\) of prime-power order, one has that the index \(|G:AC_G(A)|\) divides a prime. A nilpotent group is shown to have this property if and only if its Sylow subgroups are either Abelian or have an Abelian maximal subgroup, and the central quotient is of maximal class. Following the study of the case of groups divisible by two primes only, soluble groups with the property are also classified in ingenious terms, for whose details we refer to the paper. Finally, the insoluble case is dealt with; here the only non-Abelian composition factors that occur are some \(\text{PSL}(2,q)\), for suitable \(q\).