Difference between revisions of "Omega-1 of center is normality-large in nilpotent p-group"

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Note that if <math>G</math> is a finite p-group, i.e., a [[group of prime power order]], then it is nilpotent.
 
Note that if <math>G</math> is a finite p-group, i.e., a [[group of prime power order]], then it is nilpotent.
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==Related facts==
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===Corollaries===
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* [[Minimal normal implies contained in Omega-1 of center for nilpotent p-group]]
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* [[Socle equals Omega-1 of center for nilpotent p-group]]
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* [[Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group]]
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===Other related facts===
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* [[Minimal normal implies central in nilpotent]]
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* [[Minimal characteristic implies central in nilpotent]]
  
 
==Facts used==
 
==Facts used==
  
* [[Nilpotent implies center is normality-large]]
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# [[Nilpotent implies center is normality-large]]
* [[Omega-1 is large]] (and hence, is normality-large)
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# [[Omega-1 is large]] (and hence, is normality-large)
* [[Normality-largeness is transitive]]
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[[Normality-largeness is transitive]]

Latest revision as of 13:46, 23 September 2008

Statement

Let G be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime p. Then, the subgroup \Omega_1(Z(G)) is a normality-large subgroup of G: its intersection with every nontrivial normal subgroup is nontrivial.

Here, \Omega_1 denotes the omega subgroup: the subgroup generated by all the elements of order p, and Z(G) denotes the center of G.

Note that if G is a finite p-group, i.e., a group of prime power order, then it is nilpotent.

Related facts

Corollaries

Other related facts

Facts used

  1. Nilpotent implies center is normality-large
  2. Omega-1 is large (and hence, is normality-large)
  3. Normality-largeness is transitive