On finite groups all of whose elements of the same order are conjugate in their automorphism groups (Q1204389)

The aim of the paper under review is to study finite AT-groups (defined in the title). The main result is Theorem 3.1 that describes the general structure of arbitrary AT-groups using the classification of finite simple groups. Its statement is too long, we can mention only that a simple AT-group is isomorphic to either \(L_ 3(4)\), or \(L_ 2(q)\) for suitable \(q\in\{5,7,8,9\}\). The author considers also solvable AT-groups. In particular, it is proved that the Suzuki groups \(A(n,\theta)\) and the Sylow 2-subgroups of \(U_ 3(2^ n)\) are AT-groups. However, the question whether other Suzuki groups are AT-groups remains open.