Equivalence of definitions of nilpotency class

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This article gives a proof/explanation of the equivalence of multiple definitions for the term nilpotency class
View a complete list of pages giving proofs of equivalence of definitions

The definitions that we have to prove as equivalent

In terms of central series

The nilpotence class of a nilpotent group is the minimum possible length of a central series (where length means, the number of inequality signs, which is one less than the number of terms in the series).

In terms of lower central series

The nilpotence class of a nilpotent group is the length of its lower central series, i.e., for a group G, it is the smallest c such that G_{c+1} is trivial.

In terms of upper central series

The nilpotence class of a nilpotent group is the length of its upper central series, i.e., for a group G, it is the smallest c such that Z^c(G) = G.

Facts used

  1. Upper central series is fastest ascending central series
  2. Lower central series is fastest descending central series

Proof

We'll show the following:

Upper central series definition = Central series definition = Lower central series definition

Equality of upper central series definition and central series definition

Clearly, the upper central series, when finite, is itself a central series, so its length puts an upper bound on the minimum possible length of a central series. But the fact (fact (1)) that the upper central series is the fastest ascending central series, also shows that every central series has length at least as much as the upper central series. Thus, the length of the upper central series is the minimum possible length of a central series.

Equality of lower central series definition and central series definition

Clearly, the lower central series, when finite, is itself a central series, so its length puts an upper bound on the minimum possible length of a central series. But the fact (fact (2)) that the lower central series is the fastest descending central series, also shows that every central series has length at least as much as the lower central series. Thus, the length of the lower central series is the minimum possible length of a central series.