Abelian fully invariant subgroup: Difference between revisions
No edit summary |
|||
| Line 4: | Line 4: | ||
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''abelian fully invariant subgroup''' or '''fully invariant abelian subgroup''' if <math>H</math> is an [[abelian group]] as a group in its own right (or equivalently, is an [[abelian subgroup]] of <math>G</math>) and is also a [[fully invariant subgroup]] (or fully characteristic subgroup) of <math>G</math>, i.e., for any [[endomorphism]] <math>\sigma</math> of <math>G</math>, we have <math>\sigma(H) \subseteq H</math>. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''abelian fully invariant subgroup''' or '''fully invariant abelian subgroup''' if <math>H</math> is an [[abelian group]] as a group in its own right (or equivalently, is an [[abelian subgroup]] of <math>G</math>) and is also a [[fully invariant subgroup]] (or fully characteristic subgroup) of <math>G</math>, i.e., for any [[endomorphism]] <math>\sigma</math> of <math>G</math>, we have <math>\sigma(H) \subseteq H</math>. | ||
==Examples== | |||
{{subgroups satisfying group-subgroup property conjunction sorted by importance rank|fully invariant subgroup|abelian group}} | |||
==Facts== | ==Facts== | ||
Revision as of 20:10, 12 August 2013
This article describes a property that arises as the conjunction of a subgroup property: fully invariant subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed an abelian fully invariant subgroup or fully invariant abelian subgroup if is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of ) and is also a fully invariant subgroup (or fully characteristic subgroup) of , i.e., for any endomorphism of , we have .
Examples
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:
| Group part | Subgroup part | Quotient part | |
|---|---|---|---|
| Center of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z2 | Klein four-group |
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Facts
- Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group of nilpotency class , all the subgroups are abelian characteristic subgroups.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| fully invariant subgroup of abelian group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian characteristic subgroup | abelian and a characteristic subgroup -- invariant under all automorphisms | follows from fully invariant implies characteristic | follows from characteristic not implies fully invariant in finite abelian group | |FULL LIST, MORE INFO |
| abelian normal subgroup | abelian and a normal subgroup -- invariant under all inner automorphisms | (via abelian characteristic, follows from characteristic implies normal) | follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property | |FULL LIST, MORE INFO |
| abelian subnormal subgroup | abelian and a subnormal subgroup | |FULL LIST, MORE INFO |