Central Series - Connection Between Lower and Upper Central Series

Connection Between Lower and Upper Central Series

There are various connections between the lower central series and upper central series (Ellis 2001), particularly for nilpotent groups.

Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree (and is called the nilpotency class of the group).

However, the LCS stabilizes at the zeroth step if and only if it is perfect, while the UCS stabilizes at the zeroth step if and only if it is centerless, which are distinct concepts, and show that the lengths of the LCS and UCS need not agree in general.

For a perfect group, the UCS always stabilizes by the first step, a fact called Grün's lemma. However, a centerless group may have a very long lower central series: a noncyclic free group is centerless, but its lower central series does not stabilize until the first infinite ordinal.

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