Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me wondering:

> do (finite) groups with the property that every normal subgroup is characteristic
> have a name and/or can they be completely
> classified? Generally, has this
> property been investigated at all?

Apart from cyclic groups, some groups possessing the above property that immediately come to mind are simple groups and dihedral groups. Among 2-groups, the generalised quaternions and the [quasi-dihedral][1] groups, as well as the other group mentioned in that wikipedia article also fit the bill.

Also, every normal subgroup of a group with the above property must also have the same property. In particular the set of subnormal and that of normal subgroups must coincide. This is as far as I got on my short walk home. I can speculate about properties that the chief series of such groups must have (and maybe even conditions in terms of the chief series that are equivalent to the above), but I will rather wait for answers, since I suspect that this property might be well studied.


  [1]: http://en.wikipedia.org/wiki/Quasidihedral_group