# descending series

Let $G$ be a group.

A *descending series* of $G$
is a family ${({H}_{\alpha})}_{\alpha \le \beta}$ of subgroups^{} of $G$,
where $\beta $ is an ordinal^{},
such that ${H}_{0}=G$ and ${H}_{\beta}=\{1\}$,
and ${H}_{\alpha +1}\mathrm{\u22b4}{H}_{\alpha}$ for all $$,
and

$$ |

whenever $\delta \le \beta $ is a limit ordinal^{}.

Note that this is a generalization^{} of the concept of a subnormal series.
Compare also the dual concept of an ascending series.

Given a descending series ${({H}_{\alpha})}_{\alpha \le \beta}$,
the subgroups ${H}_{\alpha}$ are called the *terms* of the series
and the quotients (http://planetmath.org/QuotientGroup) ${H}_{\alpha}/{H}_{\alpha +1}$
are called the *factors* of the series.

A subgroup of $G$ that is a term of some descending series of $G$
is called a *descendant subgroup* of $G$.

A descending series of $G$
in which all terms are normal in $G$
is called a *descending normal series*.

Let $\U0001d51b$ be a property of groups.
A group is said to be *hypo-$\mathrm{X}$*
if it has a descending normal series
whose factors all have property $\U0001d51b$.
So, for example, a *hypoabelian group*
is a group that has a descending normal series with abelian^{} factors.
Hypoabelian groups are sometimes called *SD-groups*;
they are precisely the groups that have no non-trivial perfect subgroups,
and they are also precisely the groups
in which the transfinite derived series eventually reaches $\{1\}$.

Title | descending series |

Canonical name | DescendingSeries |

Date of creation | 2013-03-22 16:14:52 |

Last modified on | 2013-03-22 16:14:52 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 7 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E15 |

Classification | msc 20F22 |

Related topic | AscendingSeries |

Related topic | SubnormalSeries |

Related topic | SubnormalSubgroup |

Defines | descending normal series |

Defines | descendant subgroup |

Defines | descendant |

Defines | hypoabelian group |

Defines | hypoabelian |

Defines | SD-group |