# transfinite derived series

The *transfinite derived series* of a group is
an extension^{} of its derived series, defined as follows.
Let $G$ be a group and let ${G}^{(0)}=G$.
For each ordinal^{} $\alpha $
let ${G}^{(\alpha +1)}$ be the derived subgroup of ${G}^{(\alpha )}$.
For each limit ordinal^{} $\delta $
let ${G}^{(\delta )}={\bigcap}_{\alpha \in \delta}{G}^{(\alpha )}$.

Every member of the transfinite derived series of $G$ is a fully invariant subgroup of $G$.

The transfinite derived series eventually terminates, that is,
there is some ordinal $\alpha $ such that ${G}^{(\alpha +1)}={G}^{(\alpha )}$.
All remaining terms of the series are then equal to ${G}^{(\alpha )}$,
which is called the *perfect radical* or *maximum perfect subgroup*
of $G$, and is denoted $\mathcal{P}G$.
As the name suggests, $\mathcal{P}G$ is perfect,
and every perfect subgroup^{} (http://planetmath.org/Subgroup) of $G$ is a subgroup of $\mathcal{P}G$.
A group in which the perfect radical is trivial
(that is, a group without any non-trivial perfect subgroups)
is called a hypoabelian group.
For any group $G$, the quotient^{} (http://planetmath.org/QuotientGroup) $G/\mathcal{P}G$
is hypoabelian, and is sometimes called the *hypoabelianization* of $G$
(by analogy^{} with the abelianization^{}).

A group $G$ for which ${G}^{(n)}$ is trivial for some finite $n$
is called a solvable group^{}.
A group $G$ for which ${G}^{(\omega )}$ (the intersection^{} of the derived series)
is trivial is called a residually solvable group.
Free groups^{} (http://planetmath.org/FreeGroup) of rank greater than $1$
are examples of residually solvable groups that are not solvable.

Title | transfinite derived series |

Canonical name | TransfiniteDerivedSeries |

Date of creation | 2013-03-22 14:16:33 |

Last modified on | 2013-03-22 14:16:33 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 14 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F19 |

Classification | msc 20F14 |

Related topic | DerivedSubgroup |

Defines | perfect radical |

Defines | maximum perfect subgroup |

Defines | hypoabelianization |

Defines | hypoabelianisation |