Nilpotent not implies UL-equivalent

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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., nilpotent group) need not satisfy the second group property (i.e., UL-equivalent group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about nilpotent group|Get more facts about UL-equivalent group


A nilpotent group need not be UL-equivalent: its Upper central series (?) and Lower central series (?) need not coincide.

Related facts

Conditions strong enough to imply UL-equivalent

See UL-equivalent group#Stronger properties.

Lower central series versus upper central series

Here are some contrasts between facts that are true for lower central series and upper central series. Each of these pairs of facts gives further examples of nilpotent groups that are not UL-equivalent.

Nature of fact Fact for lower central series Fact for upper central series
Is the series a strongly central series? Lower central series is strongly central upper central series not is strongly central (i.e., the upper central series need not always be a strongly central series).
What is the nilpotency class of the members of the series? Second half of lower central series of nilpotent group comprises abelian groups, Penultimate term of lower central series is abelian in nilpotent group of class at least three Upper central series may be tight with respect to nilpotency class
Are the members verbal subgroups and/or fully invariant subgroups in the whole group? Lower central series members are verbal (and since verbal implies fully invariant, they are also fully invariant) Upper central series members need not be fully invariant (even for a nilpotent group)


Below are links to lists of examples:


A direct product of groups of different nilpotence classes

Suppose G,H are groups of nilpotency classes c,d respectively, with c < d. Then, the direct product G \times H is a nilpotent group.

The lower central series and upper central series are both coordinate-wise. Hence, if \gamma_i(G) denotes the i^{th} term of the lower central series with \gamma_1(G) = G, then:

\gamma_i(G \times H) = \gamma_i(G) \times \gamma_i(H)

Thus, we have:

\gamma_k(G \times H) = \{ e \} \times \gamma_k(H) \ \forall \ k \ge c+1

On the other hand, we have:

Z^{d+1-k}(G \times H) = Z^{d+1-k}(G) \times Z^{d+1-k}(H)

In particular, if c+1 \le k < d + 1, then \gamma_k(G \times H) has trivial G-projection, but Z^{d+1-k}(G \times H) does not.

Particular examples

The smallest concrete examples of the direct product construction are given below:

Direct product Order First factor Nilpotency class Second factor Nilpotency class
Direct product of D8 and Z2 16 Cyclic group:Z2 1 Dihedral group:D8 2
Direct product of Q8 and Z2 16 Cyclic group:Z2 1 Quaternion group 2
Direct product of D8 and Z3 24 Cyclic group:Z3 1 Dihedral group:D8 2
Direct product of Q8 and Z3 24 Cyclic group:Z3 1 Dihedral group:D8 2

Some examples that are not part of this general construction are nontrivial semidirect product of Z4 and Z4, central product of D8 and Z4, and holomorph of Z8.