Characteristicity does not satisfy lower central series condition

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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}

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}

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).