Powering-invariance does not satisfy lower central series condition in nilpotent group
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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It is possible to have a nilpotent group , a powering-invariant subgroup , and a positive integer such that , the member of the lower central series of , is not a powering-invariant subgroup of , the member of the lower central series of .
In fact, for any , we can construct an example that works for that value of .
Proof for the derived subgroup (i.e., )
Suppose is the direct product . The first direct factor here is unitriangular matrix group:UT(3,Q) and the second direct factor is the group of integers. Let be the subgroup inside the first direct factor. Then:
- is a powering-invariant subgroup of , because is not powered over any prime.
- is not a powering-invariant subgroup of : inside looks like Z in Q, so it is not powering-invariant.
Proof for arbitrary
Take and let inside the first direct factor. Here, denotes the unitriangular matrix group.