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

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(Proof)
(Statement)
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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.
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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>, the upper central series of <math>Z_k(G)</math> is precisely the first <math>k</math> terms of the upper central series of <math>G</math>. In particular, we can find a <math>G</math> with the property that each <math>Z_k(G)</math> has nilpotence class precisely <math>k</math>: in other words, it is not a group of class <math>k - 1</math>.  
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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 c be any natural number. Then, we can construct a nilpotent group G of nilpotency class c with the following property.

Let Z_k(G) denote the k^{th} member of the Upper central series (?) of G: Z_1(G) = Z(G) is the center and Z_k(G)/Z_{k-1}(G) is the center of G/Z_{k-1}(G) for all k. By definition of Nilpotency class (?), Z_c(G) = G.

We can find a G with the property that for any k \le c, Z_k(G) has nilpotency class precisely k.

Related facts

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

Proof

Let H_1, H_2, \dots H_c be groups such that each H_k is a nilpotent group of nilpotency class precisely k, i.e., it is not nilpotent of class smaller than k. Define G as the external direct product:

G = H_1 \times H_2 \times \dots \times H_c

Now, for each k, we have:

Z_k(G) = Z_k(H_1) \times Z_k(H_2) \times \dots \times Z_k(H_c)

In particular, we obtain that:

Z_k(G) = H_1 \times H_2 \times \dots \times H_k \times Z_k(H_{k+1}) \times \dots \times Z_k(H_c)

From the given data, in particular the fact that H_k has nilpotency class exactly k, it is clear that Z_k(G) has nilpotency class exactly k.