# ascending series

Let $G$ be a group.

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

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

A subgroup of $G$ that is a term of some ascending series of $G$
is called an *ascendant subgroup* of $G$.
The notation $H\mathrm{asc}G$ is sometimes used
to indicate that $H$ is an ascendant subgroup of $G$.

The groups in which every subgroup is ascendant
are precisely the groups that satisfy the normalizer condition.
Groups in which every cyclic subgroup is ascendant
are called *Gruenberg groups*.
It can be shown that in a Gruenberg group,
every finitely generated subgroup is ascendant and nilpotent
(and so, in particular, Gruenberg groups are locally nilpotent).

An ascending series of $G$
in which all terms are normal in $G$
is called an *ascending normal series*.

Let $\U0001d51b$ be a property of groups.
A group is said to be *hyper-$\mathrm{X}$*
if it has an ascending normal series
whose factors all have property $\U0001d51b$.
So, for example, a *hyperabelian group*
is a group that has an ascending normal series with abelian^{} factors.
Hyperabelian groups are sometimes called *$S\mathit{}{I}^{\mathrm{*}}$-groups*.

Title | ascending series |

Canonical name | AscendingSeries |

Date of creation | 2013-03-22 16:14:55 |

Last modified on | 2013-03-22 16:14:55 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E15 |

Classification | msc 20F22 |

Related topic | DescendingSeries |

Related topic | SubnormalSeries |

Related topic | SubnormalSubgroup |

Defines | ascending normal series |

Defines | ascendant subgroup |

Defines | ascendant |

Defines | hyperabelian group |

Defines | hyperabelian |

Defines | Gruenberg group |