Class two characteristic subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
A [[subgroup]] of a [[group]] is termed a '''class two characteristic | A [[subgroup]] of a [[group]] is termed a '''class two characteristic subgroup''' if it is a [[group of nilpotency class two]] and is a [[characteristic subgroup]] of the whole group. | ||
Note that ''group of nilpotency class two'' means group whose nilpotency class at ''most'' two, so this includes [[abelian characteristic subgroup]]s. | Note that ''group of nilpotency class two'' means group whose nilpotency class at ''most'' two, so this includes [[abelian characteristic subgroup]]s. | ||
==Examples== | |||
{{group-subgroup property conjunction see examples|characteristic subgroup|group of nilpotency class two}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 22:49, 2 November 2009
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): group of nilpotency class two
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a class two characteristic subgroup if it is a group of nilpotency class two and is a characteristic subgroup of the whole group.
Note that group of nilpotency class two means group whose nilpotency class at most two, so this includes abelian characteristic subgroups.
Examples
VIEW: subgroups satisfying this property | subgroups dissatisfying property characteristic subgroup | subgroups dissatisfying property group of nilpotency class two
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
Stronger properties
- Abelian characteristic subgroup
- Cyclic characteristic subgroup
- Critical subgroup
- Aut-abelian characteristic subgroup