descending series DescendingSeries 2006-09-15 16:23:53 2007-06-13 14:23:45 Definition descending normal series descendant subgroup descendant hypoabelian group hypoabelian SD-group \usepackage{amsfonts} \def\normal{\trianglelefteq} \PMlinkescapeword{factor} \PMlinkescapeword{factors} \PMlinkescapeword{property} \PMlinkescapeword{satisfy} \PMlinkescapeword{series} \PMlinkescapeword{term} \PMlinkescapeword{terms} Let $G$ be a group. A \emph{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 \emph{terms} of the series and the \PMlinkname{quotients}{QuotientGroup} $H_\alpha/H_{\alpha+1}$ are called the \emph{factors} of the series. A subgroup of $G$ that is a term of some descending series of $G$ is called a \emph{descendant subgroup} of $G$. A descending series of $G$ in which all terms are normal in $G$ is called a \emph{descending normal series}. Let $\mathfrak{X}$ be a property of groups. A group is said to be \emph{hypo-$\mathfrak{X}$} if it has a descending normal series whose factors all have property $\mathfrak{X}$. So, for example, a \emph{hypoabelian group} is a group that has a descending normal series with abelian factors. Hypoabelian groups are sometimes called \emph{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 $\{1\}$.