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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
Equivalent definitions in tabular format
No. |
Shorthand |
A subgroup of a group is termed homomorph-containing if ... |
A subgroup of a group is termed a homomorph-containing subgroup of if ...
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1 |
contains every homomorphic image |
it contains any homomorphic image of itself in the whole group. |
for any homomorphism of groups , .
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2 |
homomorphism to whole group restricts to endomorphism |
every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop. |
for any homomorphism of groups , and the restriction of to is an endomorphism of .
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3 |
(definition in terms of Hom-set maps) |
(too complicated to state without symbols) |
the natural map (by inclusion) is a surjective map of sets.
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Examples
Extreme examples
- Every group is homomorph-containing as a subgroup of itself.
- The trivial subgroup is homomorph-containing in any group.
Important classes of examples
See also the section #Stronger properties in this page.
Examples in small finite groups
Below are some examples of a proper nontrivial subgroup that satisfy the property homomorph-containing subgroup.
Below are some examples of a proper nontrivial subgroup that does not satisfy the property homomorph-containing subgroup.
Metaproperties
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Here is a summary:
Relation with other properties
Stronger properties
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
|
order-containing subgroup |
contains every subgroup whose order divides its order |
order-containing implies homomorph-containing |
homomorph-containing not implies order-containing |
Right-transitively homomorph-containing subgroup, Subhomomorph-containing subgroup|FULL LIST, MORE INFO
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subhomomorph-containing subgroup |
contains every homomorphic image of every subgroup |
subhomomorph-containing implies homomorph-containing |
homomorph-containing not implies subhomomorph-containing |
Right-transitively homomorph-containing subgroup|FULL LIST, MORE INFO
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variety-containing subgroup |
contains every subgroup of the whole group in the variety it generates |
(via subhomomorph-containing) |
(via subhomomorph-containing) |
Right-transitively homomorph-containing subgroup, Subhomomorph-containing subgroup|FULL LIST, MORE INFO
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normal Sylow subgroup |
normal and a Sylow subgroup |
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Complemented homomorph-containing subgroup, Normal Hall subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
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normal Hall subgroup |
normal and a Hall subgroup |
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Complemented homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
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fully invariant direct factor |
fully invariant and a direct factor |
equivalence of definitions of fully invariant direct factor |
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Complemented homomorph-containing subgroup, Left-transitively homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO
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left-transitively homomorph-containing subgroup |
if whole group is homomorph-containing in some group, so is the subgroup |
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homomorph-containment is not transitive |
|FULL LIST, MORE INFO
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right-transitively homomorph-containing subgroup |
any homomorph-containing subgroup of it is homomorph-containing in the whole group |
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|FULL LIST, MORE INFO
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normal subgroup having no nontrivial homomorphism to its quotient group |
no nontrivial homomorphism to the quotient group |
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|FULL LIST, MORE INFO
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Weaker properties
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
|
fully invariant subgroup |
invariant under all endomorphisms |
homomorph-containing implies fully invariant |
fully invariant not implies homomorph-containing |
Intermediately fully invariant subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
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intermediately fully invariant subgroup |
fully invariant in every intermediate subgroup |
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|FULL LIST, MORE INFO
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strictly characteristic subgroup |
invariant under all surjective endomorphisms |
(via fully invariant) |
(via fully invariant) |
Intermediately strictly characteristic subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
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characteristic subgroup |
invariant under all automorphisms |
(via fully invariant) |
(via fully invariant) |
Intermediately characteristic subgroup, Intermediately fully invariant subgroup, Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup, Isomorph-containing subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
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intermediately characteristic subgroup |
characteristic in every intermediate subgroup |
(via intermediately fully invariant) |
(via intermediately fully invariant) |
Intermediately fully invariant subgroup, Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
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normal subgroup |
invariant under all inner automorphisms, kernel of homomorphism |
(via fully invariant) |
(via fully invariant) |
Characteristic subgroup, Intermediately characteristic subgroup, Isomorph-containing subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
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isomorph-containing subgroup |
contains all isomorphic subgroups |
homomorph-containing implies isomorph-containing |
isomorph-containing not implies homomorph-containing |
|FULL LIST, MORE INFO
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homomorph-dominating subgroup |
every homomorphic image is contained in some conjugate subgroup |
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|FULL LIST, MORE INFO
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