Characteristically metacyclic not implies metacyclic derived series
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., characteristically metacyclic group) need not satisfy the second group property (i.e., group with metacyclic derived series)
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Example of the dihedral group
Further information: dihedral group:D8
Consider the dihedral group of order eight:
Let be the four-element cyclic subgroup . Then, both and are cyclic, but the abelianization is isomorphic to a Klein four-group, which is not cyclic.