Difference between revisions of "Normal of order equal to least prime divisor of group order implies central"

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==Related facts==
 
==Related facts==
  
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===Similar facts===
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* [[Normal of order two implies central]]
 
* [[Cyclic normal Sylow subgroup for least prime divisor is central]], used to show that [[cyclic Sylow subgroup for least prime divisor has normal complement]]
 
* [[Cyclic normal Sylow subgroup for least prime divisor is central]], used to show that [[cyclic Sylow subgroup for least prime divisor has normal complement]]
* [[Minimal normal implies central in nilpotent]]
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* [[Minimal normal implies central in nilpotent]], leading to [[minimal normal implies contained in Omega-1 of center for nilpotent p-group]] or equivalently [[socle equals Omega-1 of center in nilpotent p-group]]
* [[Socle equals Omega-1 of center in nilpotent p-group]]
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===Dual facts===
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* [[Subgroup of index equal to least prime divisor of group order is normal]]
 
* [[Subgroup of index equal to least prime divisor of group order is normal]]
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* [[Index two implies normal]]
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===Other normal-to-central facts===
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* [[Totally disconnected and normal in connected implies central]]
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* [[Cartan-Brauer-Hua theorem]]
  
 
==Proof==
 
==Proof==
  
 
Take any element inside the normal subgroup. The size of the conjugacy class of that element is strictly less than the least prime divisor (because the identity element is in a different conjugacy class from other elements). Since the conjugacy class of the element is an orbit of the element under the group action under conjugation, the size of the [[conjugacy class]] divides the [[order]] of the group. The only divisor of the order of the group, which is less than the least prime divisor, is 1. Thus every conjugacy class has size 1, and the subgroup is thus contained in the center.
 
Take any element inside the normal subgroup. The size of the conjugacy class of that element is strictly less than the least prime divisor (because the identity element is in a different conjugacy class from other elements). Since the conjugacy class of the element is an orbit of the element under the group action under conjugation, the size of the [[conjugacy class]] divides the [[order]] of the group. The only divisor of the order of the group, which is less than the least prime divisor, is 1. Thus every conjugacy class has size 1, and the subgroup is thus contained in the center.

Latest revision as of 17:04, 12 September 2011

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This page describes additional conditions under which a subgroup property implication can be reversed, viz a weaker subgroup property, namely Normal subgroup (?), can be made to imply a stronger subgroup property, namely central subgroup
View other subgroup property implication-reversing conditions

Statement

In a finite group, a normal subgroup whose order is the least prime divisor of the order of the group, must be a central subgroup (viz, it is contianed inside the center).

Related facts

Similar facts

Dual facts

Other normal-to-central facts

Proof

Take any element inside the normal subgroup. The size of the conjugacy class of that element is strictly less than the least prime divisor (because the identity element is in a different conjugacy class from other elements). Since the conjugacy class of the element is an orbit of the element under the group action under conjugation, the size of the conjugacy class divides the order of the group. The only divisor of the order of the group, which is less than the least prime divisor, is 1. Thus every conjugacy class has size 1, and the subgroup is thus contained in the center.