# Difference between revisions of "Abelian fully invariant subgroup"

From Groupprops

(→Stronger properties) |
(→Stronger properties) |
||

Line 21: | Line 21: | ||

! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

|- | |- | ||

− | | [[Weaker than::fully invariant subgroup of abelian group]] || || || || {{intermediate notions short|abelian fully invariant subgroup|fully invariant subgroup of abelian group}} | + | | [[Weaker than::fully invariant subgroup of abelian group]] || the whole group is an [[abelian group]] and the subgroup is a [[fully invariant subgroup]] || || || {{intermediate notions short|abelian fully invariant subgroup|fully invariant subgroup of abelian group}} |

|- | |- | ||

− | | [[Weaker than::characteristic direct factor of abelian group]] || || || || {{intermediate notions short|abelian fully invariant subgroup|characteristic direct factor of abelian group}} | + | | [[Weaker than::characteristic direct factor of abelian group]] || the whole group is an [[abelian group]] and the subgroup is a [[characteristic subgroup]] as well as a [[direct factor]] || || || {{intermediate notions short|abelian fully invariant subgroup|characteristic direct factor of abelian group}} |

+ | |- | ||

+ | | [[Weaker than::abelian verbal subgroup]] || the subgroup is abelian as well as a [[verbal subgroup]] || || || {{intermediate notions short|abelian fully invariant subgroup|abelian verbal subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::verbal subgroup of abelian group]] || the subgroup is a [[verbal subgroup]] and the whole group is an [[abelian group]] || || || {{intermediate notions short|abelian fully invariant subgroup|verbal subgroup of abelian subgroup}} | ||

|} | |} | ||

## Latest revision as of 00:33, 13 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

## Contents

## 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

### Examples based on subgroup-defining functions and series

- For a solvable group, the penultimate member of the derived series (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
- For a nilpotent group, 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.

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

Group part | Subgroup part | Quotient part | |
---|---|---|---|

A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |

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:

## 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 | the whole group is an abelian group and the subgroup is a fully invariant subgroup | |FULL LIST, MORE INFO | ||

characteristic direct factor of abelian group | the whole group is an abelian group and the subgroup is a characteristic subgroup as well as a direct factor | |FULL LIST, MORE INFO | ||

abelian verbal subgroup | the subgroup is abelian as well as a verbal subgroup | |FULL LIST, MORE INFO | ||

verbal subgroup of abelian group | the subgroup is a verbal subgroup and the whole group is an 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 | Abelian characteristic subgroup|FULL LIST, MORE INFO |

abelian subnormal subgroup | abelian and a subnormal subgroup | Abelian characteristic subgroup|FULL LIST, MORE INFO |