Finite derived subgroup not implies FZ

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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 with finite derived subgroup) need not satisfy the second group property (i.e., FZ-group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about group with finite derived subgroup|Get more facts about FZ-group

Statement

It is possible to have a group with finite derived subgroup (i.e., a group whose derived subgroup is a finite group) that is not a FZ-group -- in other words, the index of the center is finite.

Related facts

Proof

The idea is to construct an infinite extraspecial group, for instance, by taking a central product of infinitely many copies of unitriangular matrix group:UT(3,p) with all the centers identified with each other. More generally, start with any group of nilpotency class two that is not abelian, and take a restricted external central product of a countably infinite number of copies, identifying the centers of all copies (in fact, identifying any subgroup intermediate between the center and the derived subgroup is good enough).