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

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(Related facts)
 
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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==
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===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:
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* [[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]]
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===Opposite facts for upper central series===
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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:
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* [[Nilpotent of class at least three implies center of second center strictly contains center]]
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==Proof==
 
==Proof==
  

Latest revision as of 23:34, 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

Opposite facts for lower central series

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

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 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.