# Difference between revisions of "Upper central series may be tight with respect to nilpotency class"

(→Proof) |
(→Related facts) |
||

(One intermediate revision by the same user not shown) | |||

Line 1: | Line 1: | ||

==Statement== | ==Statement== | ||

− | Let <math>c</math> be any natural number. Then, we can construct a [[fact about::nilpotent group]] <math>G</math> with the following property. | + | Let <math>c</math> be any natural number. Then, we can construct a [[fact about::nilpotent group;3| ]][[nilpotent group]] <math>G</math> of [[fact about::nilpotency class;3| ]][[nilpotency class]] <math>c</math> with the following property. |

Let <math>Z_k(G)</math> denote the <math>k^{th}</math> member of the [[fact about::upper central series]] of <math>G</math>: <math>Z_1(G) = Z(G)</math> is the [[center]] and <math>Z_k(G)/Z_{k-1}(G)</math> is the center of <math>G/Z_{k-1}(G)</math> for all <math>k</math>. By definition of [[fact about::nilpotency class]], <math>Z_c(G) = G</math>. | Let <math>Z_k(G)</math> denote the <math>k^{th}</math> member of the [[fact about::upper central series]] of <math>G</math>: <math>Z_1(G) = Z(G)</math> is the [[center]] and <math>Z_k(G)/Z_{k-1}(G)</math> is the center of <math>G/Z_{k-1}(G)</math> for all <math>k</math>. By definition of [[fact about::nilpotency class]], <math>Z_c(G) = G</math>. | ||

− | We can find a <math>G</math> with the property that for any <math>k \le c</math>, | + | We can find a <math>G</math> with the property that for any <math>k \le c</math>, <math>Z_k(G)</math> has [[nilpotency class]] precisely <math>k</math>. |

==Related facts== | ==Related facts== | ||

+ | |||

+ | ===Opposite facts for lower central series=== | ||

The corresponding statement is not true for the lower central series. Some related facts: | The corresponding statement is not true for the lower central series. Some related facts: | ||

Line 14: | Line 16: | ||

* [[Second half of lower central series of nilpotent group comprises abelian groups]] | * [[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]] | * [[Penultimate term of lower central series is abelian in nilpotent group of class at least three]] | ||

− | + | ||

+ | ===Opposite facts for upper central series=== | ||

+ | |||

+ | It is also true that the upper central series for any member of the upper central series (beyond the [[center]]) grows faster than the actual upper central series of the whole group. See: | ||

+ | |||

+ | * [[Nilpotent of class at least three implies center of second center strictly contains center]] | ||

+ | |||

==Proof== | ==Proof== | ||

## Latest revision as of 23:34, 2 February 2012

## Contents

## Statement

Let be any natural number. Then, we can construct a nilpotent group of nilpotency class with the following property.

Let denote the member of the Upper central series (?) of : is the center and is the center of for all . By definition of Nilpotency class (?), .

We can find a with the property that for any , has nilpotency class precisely .

## Related facts

### Opposite facts for lower central series

The corresponding statement is not true for the lower central series. Some related facts:

- Lower central series is strongly central
- 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

### Opposite facts for upper central series

It is also true that the upper central series for any member of the upper central series (beyond the center) grows faster than the actual upper central series of the whole group. See:

## Proof

Let be groups such that each is a nilpotent group of nilpotency class precisely , i.e., it is *not* nilpotent of class smaller than . Define as the external direct product:

Now, for each , we have:

In particular, we obtain that:

From the given data, in particular the fact that has nilpotency class exactly , it is clear that has nilpotency class exactly .