# descending series

Let $G$ be a group.

A descending series of $G$ is a family $(H_{\alpha})_{\alpha\leq\beta}$ of subgroups of $G$, where $\beta$ is an ordinal, such that $H_{0}=G$ and $H_{\beta}=\{1\}$, and $H_{\alpha+1}\trianglelefteq H_{\alpha}$ for all $\alpha<\beta$, and

 $\bigcap_{\alpha<\delta}H_{\alpha}=H_{\delta}$

whenever $\delta\leq\beta$ 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 $(H_{\alpha})_{\alpha\leq\beta}$, 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 $\mathfrak{X}$ be a property of groups. A group is said to be hypo-$\mathfrak{X}$ if it has a descending normal series whose factors all have property $\mathfrak{X}$. 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