A scheme of construction of infinite groups, other than simple groups, free groups of infinite rank and the infinite cyclic group, which are isomorphic to all their non-trivial normal subgroups is presented. Some results about the automorphism groups of simple infinite groups are also obtained. In particular, it is proved that there is an infinite group $G$ of any sufficiently large prime exponent $p$ (or which is torsion-free) all of whose proper subgroups are cyclic, and such that the groups ${\rm Aut}\,G$ and ${\rm Out}\,G$ are isomorphic. The proofs use the technique of graded diagrams developed by A. Yu. Ol'shanskii.

1991 Mathematics Subject Classification: 20F05, 20F06.