Lower central series not is strongly characteristic

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Template:Ssdf property dissatisfaction

Statement

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.

Note that the lower central series is strongly characteristic for an Abelian group or for a group of nilpotence class two.

Proof

Example of a maximal class group

Let p be an odd prime. Consider the wreath product of the cyclic group of order p with itself under the regular action. Equivalently, if C_p denotes the cyclic group of order p, the group is given by:

(C_p \times C_p \times \dots C_p) \rtimes C_p

where the action is by cyclic permutation of coordinates. This is a group of order p^{p + 1}, and is a maximal class group: it has class p. All the members of the lower central series, barring the first one, live inside the subgroup C_p \times C_p \times \dots C_p.

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 C_p \times C_p \times \dots C_p is elementary Abelian and hence characteristically simple. Thus, since p is an odd prime, we can find two such members and establish that the lower central series is not strongly characteristic.

Note that for p = 2, 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.