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

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

− | + | Let <math>H_1, H_2, \dots H_c</math> be [[group]]s such that each <math>H_k</math> is a [[nilpotent group]] of [[nilpotency class]] precisely <math>k</math>, i.e., it is ''not'' nilpotent of class smaller than <math>k</math>. Define <math>G</math> as the [[external direct product]]: | |

− | + | <math>G = H_1 \times H_2 \times \dots \times H_c</math> | |

− | + | Now, for each <math>k</math>, we have: | |

+ | |||

+ | <math>Z_k(G) = Z_k(H_1) \times Z_k(H_2) \times \dots \times Z_k(H_c)</math> | ||

+ | |||

+ | In particular, we obtain that: | ||

+ | |||

+ | <math>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)</math> | ||

+ | |||

+ | From the given data, in particular the fact that <math>H_k</math> has nilpotency class exactly <math>k</math>, it is clear that <math>Z_k(G)</math> has nilpotency class exactly <math>k</math>. |

## Revision as of 23:19, 2 February 2012

## Statement

Let be any natural number. Then, we can construct a Nilpotent group (?) 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 , the upper central series of is precisely the first terms of the upper central series of . In particular, we can find a with the property that each has nilpotence class precisely : in other words, it is not a group of class .

## 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 .