Lower central series not is strongly characteristic
The lower central series of a nilpotent group need not be a strongly characteristic series. In other words, it is not necessary that for any two members of the series, the smaller one is always a characteristic subgroup of the bigger one.
Example of a maximal class group
Let be an odd prime. Consider the wreath product of the cyclic group of order with itself under the regular action. Equivalently, if denotes the cyclic group of order , the group is given by:
where the action is by cyclic permutation of coordinates. This is a group of order , and is a maximal class group: it has class . All the members of the lower central series, barring the first one, live inside the subgroup .
It is easily seen that given two distinct members of the lower central series other than the whole group and the trivial subgroup, the smaller one is not characteristic in the bigger one, because any subgroup of is elementary Abelian and hence characteristically simple. Thus, since is an odd prime, we can find two such members and establish that the lower central series is not strongly characteristic.
Note that for , the problem is that the lower central series is too short for us to be able to find two distinct members neither of which is the trivial subgroup or the whole group.