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 $H_{\alpha+1}\normal H_\alpha$ for all $\alpha<\beta$ and $$\bigcap_{\alpha<\delta}H_\alpha=H_\delta$$ 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 $H_\alpha/H_{\alpha+1}$ 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 <</SPAN>#69#>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 $\{1\}$