Powering-invariance does not satisfy lower central series condition in nilpotent group

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This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., lower central series condition)
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., powering-invariant subgroup) need not satisfy the second subgroup property (i.e., LCS-powering-invariant subgroup)
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Statement

It is possible to have a nilpotent group G, a powering-invariant subgroup H, and a positive integer k such that \gamma_k(H), the k^{th} member of the lower central series of H, is not a powering-invariant subgroup of \gamma_k(G), the k^{th} member of the lower central series of G.

In fact, for any k > 1, we can construct an example that works for that value of k.

Proof

Proof for the derived subgroup (i.e., k = 2)

Suppose G is the direct product UT(3,\mathbb{Q}) \times \mathbb{Z}. The first direct factor here is unitriangular matrix group:UT(3,Q) and the second direct factor is the group of integers. Let H be the subgroup UT(3,\mathbb{Z}) inside the first direct factor. Then:

  • H is a powering-invariant subgroup of G, because G is not powered over any prime.
  • H' is not a powering-invariant subgroup of G': H' inside G' looks like Z in Q, so it is not powering-invariant.

Proof for arbitrary k

Take G = UT(k+1,\mathbb{Q}) \times \mathbb{Z} and let H = UT(k+1,\mathbb{Z}) inside the first direct factor. Here, UT denotes the unitriangular matrix group.