# Changes

## Hypoabelian group

, 14:35, 24 October 2009
no edit summary
{{group property}}
{{variationofvariation of|solvabilitysolvable group}} {{oppositeofopposite of|perfectnessperfect group}}
==Definition==
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==
===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 properties===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}}|-* | [[Locally solvable Weaker than::Free group]]|| Free on a given generating set || (via residually solvable) || (via residually solvable) || {{intermediate notions short|hypoabelian group|free group}}|}