# Difference between revisions of "Hypoabelian group"

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{{group property}} | {{group property}} | ||

− | {{ | + | {{variation of|solvable group}} |

− | + | {{opposite of|perfect group}} | |

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==Definition== | ==Definition== | ||

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A [[group]] is termed '''hypoabelian''' if the following equivalent conditions are satisfied: | A [[group]] is termed '''hypoabelian''' if the following equivalent conditions are satisfied: | ||

− | + | # The [[defining ingredient::perfect core]] is [[trivial group|trivial]] | |

− | + | # The [[defining ingredient::hypoabelianization]] is the quotient by the trivial subgroup, and hence, isomorphic to the whole group. | |

− | + | # The transfinite [[derived series]] terminates at the identity. (Note that this is the ''transfinite'' derived series, where the successor of a given subgroup is its [[commutator subgroup]] and subgroups at limit ordinals are given by intersecting all previous subgroups.) | |

− | + | # There is no nontrivial [[perfect group|perfect]] subgroup. | |

− | + | # There is a descending transfinite [[normal series]] where all the successive quotients are abelian | |

==Relation with other properties== | ==Relation with other properties== | ||

Line 21: | Line 19: | ||

===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="wikitable" border="1" | |

− | + | ! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |

− | + | |- | |

− | + | | [[Weaker than::Solvable group]] || [[derived series]] terminates at identity in finitely many steps || [[solvable implies hypoabelian]] || [[hypoabelian not implies solvable]] || {{intermediate notions short|hypoabelian group|solvable group}} | |

− | + | |- | |

− | + | | [[Weaker than::Hypocentral group]] || transfinite [[lower central series]] terminates at identity || [[hypocentral implies hypoabelian]] || [[hypoabelian not implies hypocentral]] || {{intermediate notions short|hypoabelian group|hypocentral group}} | |

+ | |- | ||

+ | | [[Weaker than::Residually solvable group]] || intersection of all finite members of [[derived series]] is identity || [[residually solvable implies hypoabelian]] || [[hypoabelian not implies residually solvable]] || {{intermediate notions short|hypoabelian group|residually solvable group}} | ||

+ | |- | ||

+ | | [[Weaker than::Free group]] || Free on a given generating set || (via residually solvable) || (via residually solvable) || {{intermediate notions short|hypoabelian group|free group}} | ||

+ | |} |

## Latest revision as of 14:35, 24 October 2009

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of solvable group|Find other variations of solvable group |

*This is an opposite of perfect group*

## Contents

## Definition

### Symbol-free definition

A group is termed **hypoabelian** if the following equivalent conditions are satisfied:

- The perfect core is trivial
- The hypoabelianization is the quotient by the trivial subgroup, and hence, isomorphic to the whole group.
- The transfinite derived series terminates at the identity. (Note that this is the
*transfinite*derived series, where the successor of a given subgroup is its commutator subgroup and subgroups at limit ordinals are given by intersecting all previous subgroups.) - There is no nontrivial perfect subgroup.
- There is a descending transfinite normal series where all the successive quotients are abelian

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Solvable group | derived series terminates at identity in finitely many steps | solvable implies hypoabelian | hypoabelian not implies solvable | Residually solvable group|FULL LIST, MORE INFO |

Hypocentral group | transfinite lower central series terminates at identity | hypocentral implies hypoabelian | hypoabelian not implies hypocentral | |FULL LIST, MORE INFO |

Residually solvable group | intersection of all finite members of derived series is identity | residually solvable implies hypoabelian | hypoabelian not implies residually solvable | |FULL LIST, MORE INFO |

Free group | Free on a given generating set | (via residually solvable) | (via residually solvable) | Residually nilpotent group, Residually solvable group|FULL LIST, MORE INFO |