# Equivalence of definitions of nilpotency class

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 , it is the smallest such that 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 , it is the smallest such that .

## Facts used

- Upper central series is fastest ascending central series
- 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.