Subnormal series
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This article defines a property that can be evaluated for a subgroup seriesView a complete list of properties of subgroup series
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition
Definition for finite length
A subnormal series is a subgroup series where each member of the series is normal in the next one containing it. In symbols:
- A descending series:
of subgroups of a group is termed a subnormal series if
is a normal subgroup of
for
.
- An ascending series:
of subgroups of a group is termed a subnormal series if each
is a normal subgroup of
.
Note that the subnormal series must have its largest member equal to the whole group. In some contexts, the term subnormal series refers to a subnormal series that terminates at the trivial subgroup. Note that any subnormal series of a group can be extended to such a subnormal series by adding the trivial group at the end.
Definition for infinite length
Further information: Subnormal series of infinite length
Relation with other properties
Stronger properties
Related subgroup properties
- A subnormal subgroup is a subgroup for which there is a subnormal series of finite length starting at the subgroup and ending at the whole group.
- An ascendant subgroup is a subgroup for which there is an ascending subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.
- A descendant subgroup is a subgroup for which there is a descending subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.
- A serial subgroup is a subgroup for which there is a subnormal series of possibly infinite length starting at the subgroup and ending at the whole group.