Normalizer condition 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., group satisfying normalizer condition) need not satisfy the second group property (i.e., nilpotent group)
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It is possible to have a group G that lsatisfies the normailzer condition, i.e., it has no proper self-normalizing subgroup, but G itself is not nilpotent.


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.