Maximal among abelian normal subgroups: Difference between revisions
(New page: {{subgroup property}} ==Definition== ===Symbol-free definition=== A subgroup of a group is termed '''maximal among Abelian normal subgroups''' if it is an [[Abelian normal subgr...) |
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''maximal among Abelian normal subgroups''' if <math>H</math> is an [[Abelian normal subgroup]] of <math>G</math>, and for any <math>K</math> containing <math>H</math> that is an Abelian normal subgroup of <math>G</math>, <math>H = K</math>. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''maximal among Abelian normal subgroups''' if <math>H</math> is an [[Abelian normal subgroup]] of <math>G</math>, and for any <math>K</math> containing <math>H</math> that is an Abelian normal subgroup of <math>G</math>, <math>H = K</math>. | ||
==Formalisms== | |||
{{obtainedbyapplyingthe|maximal operator|Abelian normal subgroup}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 17:57, 8 February 2008
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]
Definition
Symbol-free definition
A subgroup of a group is termed maximal among Abelian normal subgroups if it is an Abelian normal subgroup and there is no Abelian normal subgroup properly containing it.
Definition with symbols
A subgroup of a group is termed maximal among Abelian normal subgroups if is an Abelian normal subgroup of , and for any containing that is an Abelian normal subgroup of , .
Formalisms
In terms of the maximal operator
This property is obtained by applying the maximal operator to the property: Abelian normal subgroup
View other properties obtained by applying the maximal operator