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