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subnormal series (Definition)

Let $ G$ be a group with a subgroup $ H$, and let

$\displaystyle G = G_0 \rhd G_1 \rhd \cdots \rhd G_n = H$ (1)

be a series of subgroups with each $ G_i$ a normal subgroup of $ G_{i-1}$. Such a series is called a subnormal series or a subinvariant series.

If in addition, each $ G_i$ is a normal subgroup of $ G$, then the series is called a normal series.

A subnormal series in which each $ G_i$ is a maximal normal subgroup of $ G_{i-1}$ is called a composition series.

A normal series in which $ G_i$ is a maximal normal subgroup of $ G$ contained in $ G_{i-1}$ is called a principal series or a chief series.

Note that a composition series need not end in the trivial group $ 1$. One speaks of a composition series (1) as a composition series from $ G$ to $ H$. But the term composition series for $ G$ generally means a composition series from $ G$ to $ 1$.

Similar remarks apply to principal series.

Some authors use normal series as a synonym for subnormal series. This usage is, of course, not compatible with the stronger definition of normal series given above.



"subnormal series" is owned by mclase.
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See Also: subnormal subgroup, Jordan-Hölder decomposition theorem, solvable group, descending series, ascending series

Other names:  subinvariant series
Also defines:  composition series, normal series, principal series, chief series
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Cross-references: similar, contained, maximal normal subgroup, normal subgroup, subgroup, group
There are 8 references to this entry.

This is version 5 of subnormal series, born on 2003-10-04, modified 2007-04-03.
Object id is 4750, canonical name is SubnormalSeries.
Accessed 7903 times total.

Classification:
AMS MSC: 20D30 (Group theory and generalizations :: Abstract finite groups :: Series and lattices of subgroups)
Pending Errata and Addenda
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