Fully invariant direct factor: Difference between revisions

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{{further|[[Equivalence of definitions of fully invariant direct factor]]}}
{{further|[[Equivalence of definitions of fully invariant direct factor]]}}


==Examples==
===Extreme examples===
* Every group is a fully invariant direct factor in itself.
* The trivial subgroup is a fully invariant direct factor in every group.
===Subgroups satisfying the property===
{{subgroups satisfying property conjunction sorted by importance rank|fully invariant subgroup|direct factor}}
===Subgroups dissatisfying the property===
{{subgroups dissatisfying property conjunction sorted by importance rank|fully invariant subgroup|direct factor}}
==Relation with other properties==
==Relation with other properties==


===Stronger properties===
===Stronger properties===


* [[Weaker than::Sylow direct factor]]
{| class="sortable" border="1"
* [[Weaker than::Hall direct factor]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::Sylow direct factor]] || [[Sylow subgroup]] that is also a [[direct factor]] || || ||
|-
| [[Weaker than::Hall direct factor]] || [[Hall subgroup]] (order and index are relatively prime to each other) that is also a [[direct factor]] || || ||
|-
| [[Weaker than::characteristic direct factor of abelian group]] || the whole group is an [[abelian group]] || || || {{intermediate notions short|fully invariant direct factor|characteristic direct factor of abelian group}}
|-
| [[Weaker than::characteristic direct factor of nilpotent group]] || the whole group is a [[nilpotent group]] || || || {{intermediate notions short|fully invariant direct factor|characteristic direct factor of nilpotent group}}
|}


===Weaker properties===
===Weaker properties===


* [[Stronger than::Normal subgroup having no nontrivial homomorphism to its quotient group]]
{| class="sortable" border="1"
* [[Stronger than::Quotient-subisomorph-containing subgroup]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Stronger than::Left-transitively homomorph-containing subgroup]]: {{proofat|[[Fully invariant direct factor implies left-transitively homomorph-containing]]}}
|-
* [[Stronger than::Homomorph-containing subgroup]]
| [[Stronger than::normal subgroup having no nontrivial homomorphism to its quotient group]] || [[normal subgroup]] such that the only homomorphism from it to its corresponding [[quotient group]] is the trivial homomorphism. || || || {{intermediate notions short|normal subgroup having no nontrivial homomorphism to its quotient group|fully invariant direct factor}}
* [[Stronger than::Fully invariant subgroup]]
|-
* [[Stronger than::Isomorph-containing subgroup]]
| [[Stronger than::quotient-subisomorph-containing subgroup]] || || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|fully invariant direct factor}}
* [[Stronger than::Characteristic direct factor]]
|-
* [[Stronger than::Characteristic subgroup]]
| [[Stronger than::left-transitively homomorph-containing subgroup]] || || [[fully invariant direct factor implies left-transitively homomorph-containing]] || || {{intermediate notions short|left-transitively homomorph-containing subgroup|fully invariant direct factor}}
* [[Stronger than::Direct factor]]
|-
| [[Stronger than::homomorph-containing subgroup]] || contains every homomorphic image of itself in the whole group || follows from [[equivalence of definitions of fully invariant direct factor]] || || {{intermediate notions short|homomorph-containing subgroup|fully invariant direct factor}}
|-
| [[Stronger than::fully invariant subgroup]] || invariant under any [[endomorphism]] of the whole group || (by definition) || [[fully invariant not implies direct factor]] || {{intermediate notions short|fully invariant subgroup|fully invariant direct factor}}
|-
| [[Stronger than::isomorph-containing subgroup]] || contains any subgroup of the whole group that is isomorphic to it. || follows from [[equivalence of definitions of fully invariant direct factor]] || || {{intermediate notions short|isomorph-containing subgroup|fully invariant direct factor}}
|-
| [[Stronger than::characteristic direct factor]] || both a [[characteristic subgroup]] and a [[direct factor]] of the whole group. || follows from [[fully invariant implies characteristic]] || [[characteristic direct factor not implies fully invariant]] || {{intermediate notions short|characteristic direct factor|fully invariant direct factor}}
|-
| [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s of the whole group. || follows from [[fully invariant implies characteristic]] || (via characteristic direct factor, or via fully invariant subgroup) || {{intermediate notions short|characteristic subgroup|fully invariant direct factor}}
|-
| [[Stronger than::direct factor]] || normal subgroup with a normal complement || by definition || follows from [[direct factor not implies characteristic]] || {{intermediate notions short|direct factor|fully invariant direct factor}}
|}

Latest revision as of 20:01, 14 July 2013

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: fully invariant subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a fully invariant direct factor if it satisfies the following equivalent conditions:

  1. It is both a fully invariant subgroup and a direct factor.
  2. It is both a homomorph-containing subgroup and a direct factor.
  3. It is both an isomorph-containing subgroup and a direct factor.
  4. It is both a quotient-subisomorph-containing subgroup and a direct factor.
  5. It is both a normal subgroup having no nontrivial homomorphism to its quotient group and a direct factor.

Equivalence of definitions

Further information: Equivalence of definitions of fully invariant direct factor

Examples

Extreme examples

  • Every group is a fully invariant direct factor in itself.
  • The trivial subgroup is a fully invariant direct factor in every group.

Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:


Here are some examples of subgroups in relatively less basic/important groups satisfying the property:


Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:


Subgroups dissatisfying the property

Template:Subgroups dissatisfying property conjunction sorted by importance rank

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Sylow direct factor Sylow subgroup that is also a direct factor
Hall direct factor Hall subgroup (order and index are relatively prime to each other) that is also a direct factor
characteristic direct factor of abelian group the whole group is an abelian group |FULL LIST, MORE INFO
characteristic direct factor of nilpotent group the whole group is a nilpotent group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup having no nontrivial homomorphism to its quotient group normal subgroup such that the only homomorphism from it to its corresponding quotient group is the trivial homomorphism. |FULL LIST, MORE INFO
quotient-subisomorph-containing subgroup |FULL LIST, MORE INFO
left-transitively homomorph-containing subgroup fully invariant direct factor implies left-transitively homomorph-containing |FULL LIST, MORE INFO
homomorph-containing subgroup contains every homomorphic image of itself in the whole group follows from equivalence of definitions of fully invariant direct factor |FULL LIST, MORE INFO
fully invariant subgroup invariant under any endomorphism of the whole group (by definition) fully invariant not implies direct factor |FULL LIST, MORE INFO
isomorph-containing subgroup contains any subgroup of the whole group that is isomorphic to it. follows from equivalence of definitions of fully invariant direct factor |FULL LIST, MORE INFO
characteristic direct factor both a characteristic subgroup and a direct factor of the whole group. follows from fully invariant implies characteristic characteristic direct factor not implies fully invariant |FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms of the whole group. follows from fully invariant implies characteristic (via characteristic direct factor, or via fully invariant subgroup) |FULL LIST, MORE INFO
direct factor normal subgroup with a normal complement by definition follows from direct factor not implies characteristic |FULL LIST, MORE INFO
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