Characteristic subgroup of finite group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[subgroup]] of a [[group]] is termed a '''characteristic subgroup of finite group''' if the whole group is a [[finite group]] and the subgroup is a [[characteristic subgroup]] of it. | A [[subgroup]] of a [[group]] is termed a '''characteristic subgroup of finite group''' if it satisfies the following equivalent conditions: | ||
# The whole group is a [[finite group]] and the subgroup is a [[characteristic subgroup]] of it. | |||
# The whole group is a [[finite group]] and the subgroup is a [[conjunction involving::strictly characteristic subgroup]][[defining ingredient::strictly characteristic subgroup| ]](i.e., invariant under all surjective endomorphisms) of it. | |||
# The whole group is a [[finite group]] and the subgroup is an [[conjunction involving::injective endomorphism-invariant subgroup]][[defining ingredient::injective endomorphism-invariant subgroup| ]] of it. | |||
==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | |||
* [[Weaker than::Isomorph-free subgroup of finite group]] | |||
* [[Weaker than::Fully invariant subgroup of finite group]] | |||
* [[Weaker than::Normal Sylow subgroup]] | |||
* [[Weaker than::Normal Hall subgroup]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
Revision as of 02:07, 13 August 2009
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: finite group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
A subgroup of a group is termed a characteristic subgroup of finite group if it satisfies the following equivalent conditions:
- The whole group is a finite group and the subgroup is a characteristic subgroup of it.
- The whole group is a finite group and the subgroup is a strictly characteristic subgroup(i.e., invariant under all surjective endomorphisms) of it.
- The whole group is a finite group and the subgroup is an injective endomorphism-invariant subgroup of it.
Relation with other properties
Stronger properties
- Isomorph-free subgroup of finite group
- Fully invariant subgroup of finite group
- Normal Sylow subgroup
- Normal Hall subgroup