Characteristicity does not satisfy lower central series condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., lower central series condition).
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Statement

It is possible to have a group $G$ and a characteristic subgroup $H$ of $G$ such that there is a positive integer $k$ for which the lower central series member $\gamma _{k}(H)$ is not a characteristic subgroup of the lower central series member $\gamma _{k}(G)$.

In fact, we can construct, for each positive integer $k>1$, an example that works for that $k$. In the special case that $k=2$, we obtain an example of a group $G$ and a characteristic subgroup $H$ of $G$ such that the derived subgroup $H'$ is not a characteristic subgroup of the derived subgroup $G'$.

Proof

Example for the derived subgroup ($k=2$)

Smallest order example for a 2-group

Further information: faithful semidirect product of E8 and Z4

Suppose $G$ is the faithful semidirect product of E8 and Z4. We can think of it as the semidirect product of elementary abelian group:E8 (a three-dimensional vector space over field:F2) and cyclic group:Z4, where the generator of the latter acts as the following matrix on the former:

${\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\\\end{pmatrix}}$

• $\gamma _{2}(G)=G'$, the derived subgroup of $G$, is a Klein four-group inside the base of the semidirect product, generated by the first two basis vectors.
• Let $H=C_{G}(G')$. $H$ contains the base of the semidirect product as well as the square of the generator of cyclic group:Z4. Explicitly, $H$ is the semidirect product of the base by the element acting as follows:

${\begin{pmatrix}1&0&1\\0&1&0\\0&0&1\\\end{pmatrix}}$

• $H$ is isomorphic to direct product of D8 and Z2. The derived subgroup $\gamma _{2}(H)=H'$ of $H$ is isomorphic to cyclic group:Z2 (generated by the second basis vector) and it lives inside $G'$ as Z2 in V4. Thus, $H'$ is not a characteristic subgroup of $G'$.

Smallest order example for a 3-group

Further information: wreath product of Z3 and Z3

Let $G$ be the group $\mathbb {Z} _{3}\wr \mathbb {Z} _{3}$. Explicitly, $G$ is the semidirect product of elementary abelian group:E27, viewed as a three-dimensional vector space over field:F3, with the acting element of order three acting by the matrix:

${\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\\\end{pmatrix}}$

• $\gamma _{2}(G)=G'$ is the subgroup in the base comprising the elements whose coordinates add up to zero. It is an elementary abelian group of prime-square order.
• $H$ is defined as the internal semidirect product of $G'$ and any element of three outside the base. There is a unique possibility for such a subgroup $H$, and it is isomorphic to unitriangular matrix group:UT(3,3).
• $\gamma _{2}(H)=H'$ is a subgroup of order 3. It lives as a group of order 3 inside $G'$ which is elementary abelian of order 9, hence is not characteristic in $G'$.

Example for larger $k$

Further information: wreath product of groups of prime order

For a positive integer $k$, choose $p$ to be a prime number greater than $k$.

• Define $G$ as the wreath product of groups of prime order $\mathbb {Z} _{p}\wr \mathbb {Z} _{p}$. This is a group of order $p^{p+1}$ with an elementary abelian normal subgroup of order $p^{p}$ and a complement of order $p$ acting by cyclic permutation of the coordinates.
• Define $H$ as the internal semidirect product of $G'$ with an element of order $p$ outside the elementary abelian normal subgroup.
• $\gamma _{k}(H)$ and $\gamma _{k}(G)$ are both elementary abelian subgroups of $G$, with $\gamma _{k}(H)$ having order $p^{p-k}$ and $\gamma _{k}(G)$ having order $p^{p-k+1}$. In particular, $\gamma _{k}(H)$ is a proper nontrivial subgroup of the elementary abelian $p$-group $\gamma _{k}(G)$, hence it is not characteristic inside $\gamma _{k}(G)$.