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

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==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== |

## Revision as of 23:21, 2 February 2012

## 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

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

## 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 .