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 $H_{\alpha +1}\mathrm{⊴}{H}_{\alpha }$ 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 seriesand 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 $𝔛$ be a property of groups.A group is said to be hypo-$\mathrm{X}$if it has a descending normal serieswhose factors all have property $𝔛$.So, for example, a hypoabelian groupis 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 groupsin which the transfinite derived series eventually reaches $\{1\}$.