Locally nilpotent not implies nilpotent

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., locally nilpotent group) need not satisfy the second group property (i.e., nilpotent group)
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It is possible to have a group G that is locally nilpotent, i.e., every finitely generated subgroup of it is nilpotent, but G itself is not nilpotent.


Example that satisfies the normalizer condition

The generalized dihedral group for 2-quasicyclic group is an example. This is obtained by taking the 2-quasicyclic group (the group of all 2^{n^{th}} roots of unity for arbitrary nonnegative integers n) and then constructing the generalized dihedral group for it: the semidirect product with the automorphism that sends each element to its negative.

This is a group satisfying normalizer condition.

Example that does not satisfy the normalizer condition

Further information: locally nilpotent not implies normalizer condition

McLain's group over a field (a unitriangular matrix group with indexing set the rationals) gives an example of a locally nilpotent group that does not satisfy the normalizer condition.