Difference between revisions of "Intersection of subnormal subgroups"

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(New page: {{subgroup property}} ==Definition== A subgroup of a group is termed an '''intersection of subnormal subgroups''' if it satisfies the following equivalent conditions: * It can b...)
 
 
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{{subgroup property}}
 
{{subgroup property}}
 
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{{finitarily equivalent to|subnormal subgroup}}
 
==Definition==
 
==Definition==
  
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* It can be expressed as an intersection of subnormal subgroups.
 
* It can be expressed as an intersection of subnormal subgroups.
  
Note that since [[subnormality is finite-intersection-closed]], and moreover, subnormality of fixed depth is closed under arbitrary intersections, these two definitions are equivalent.
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Note that since [[subnormality of fixed depth is strongly intersection-closed|subnormality of fixed depth is closed under arbitrary intersections]], these two definitions are equivalent.
  
 
==Relation with other properties==
 
==Relation with other properties==

Latest revision as of 22:12, 22 February 2009

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

Definition

A subgroup of a group is termed an intersection of subnormal subgroups if it satisfies the following equivalent conditions:

  • It can be expressed as the intersection of a descending chain of subnormal subgroups.
  • It can be expressed as an intersection of subnormal subgroups.

Note that since subnormality of fixed depth is closed under arbitrary intersections, these two definitions are equivalent.

Relation with other properties

Stronger properties

Weaker properties