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descending series (Definition)

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 $H_{\alpha+1}\trianglelefteq H_\alpha$ for all $\alpha<\beta$, and

\begin{displaymath}\bigcap_{\alpha<\delta}H_\alpha=H_\delta\end{displaymath}

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 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 $\{1\}$.



"descending series" is owned by yark.
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See Also: ascending series, subnormal series, subnormal subgroup

Also defines:  descending normal series, descendant subgroup, descendant, hypoabelian group, hypoabelian, SD-group
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Cross-references: eventually, transfinite derived series, perfect, abelian, normal, ascending series, subnormal series, limit ordinal, ordinal, subgroups, group
There are 6 references to this entry.

This is version 4 of descending series, born on 2006-09-15, modified 2007-06-13.
Object id is 8352, canonical name is DescendingSeries.
Accessed 2520 times total.

Classification:
AMS MSC: 20E15 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Chains and lattices of subgroups, subnormal subgroups)
 20F22 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Other classes of groups defined by subgroup chains)
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