Finite groups all of whose Abelian subgroups are QTI-subgroups. (Q958644)

A subgroup \(H\) of a finite group \(G\) is called a QTI-subgroup if \(C_G(x)\leq N_G(H)\) for all \(x\in H^\#\). In this note the authors classify the AQTI-groups, i.e., groups all of whose Abelian subgroups are QTI-groups. For example, if \(G\) is a non-Dedekindian \(p\)-group, then it is either \(Q_8*D_8\) (central product) of order \(2^5\), or \(|G/Z(G)|=p^2\) and \(G'\) is the unique minimal normal subgroup of \(G\).