Difference between revisions of "Characteristic subgroup of finite group"

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(Definition)
(Relation with other properties)
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! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 
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| [[Weaker than::Isomorph-free subgroup of finite group]] || || || ||
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| [[Weaker than::isomorph-free subgroup of finite group]] || || || ||
 
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| [[Weaker than::Fully invariant subgroup of finite group]] || || || ||
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| [[Weaker than::fully invariant subgroup of finite group]] || || || ||
 
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| [[Weaker than::Normal Sylow subgroup]] || || || ||
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| [[Weaker than::normal Sylow subgroup]] || || || ||
 
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| [[Weaker than::Normal Hall subgroup]] || || || ||
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| [[Weaker than::normal Hall subgroup]] || || || ||
 
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| [[Weaker than::Characteristic subgroup of group of prime power order]] || || || ||
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| [[Weaker than::xharacteristic subgroup of group of prime power order]] || || || ||
 
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! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 
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| [[Stronger than::Normal subgroup of finite group]] || || || ||
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| [[Stronger than::normal subgroup of finite group]] || || || ||
 
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| [[Stronger than::Finite characteristic subgroup]] || || || ||
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| [[Stronger than::finite characteristic subgroup]] || || || ||
 
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| [[Stronger than::Finite normal subgroup]] || || || ||
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| [[Stronger than::finite normal subgroup]] || || || ||
 
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Revision as of 14:31, 1 June 2020

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:

  1. The whole group is a finite group and the subgroup is a characteristic subgroup of it.
  2. The whole group is a finite group and the subgroup is a strictly characteristic subgroup (i.e., invariant under all surjective endomorphisms) of it.
  3. The whole group is a finite group and the subgroup is an injective endomorphism-invariant subgroup of it.
  4. The whole group is a finite group and the subgroup is a purely definable subgroup of it, i.e,, the subgroup is definable in the first-order theory of the group.
  5. The whole group is a finite group and the subgroup is an elementarily characteristic subgroup of it, i.e., there is no other elementarily equivalently embedded subgroup.
  6. The whole group is a finite group and the subgroup is a monadic second-order characteristic subgroup of it.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
isomorph-free subgroup of finite group
fully invariant subgroup of finite group
normal Sylow subgroup
normal Hall subgroup
xharacteristic subgroup of group of prime power order

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
normal subgroup of finite group
finite characteristic subgroup
finite normal subgroup