Descendant subgroup
From Groupprops
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |
This is a variation of subnormality|Find other variations of subnormality |
Contents
Definition
A subgroup of a group
is termed descendant if we have subgroups
of
for every ordinal
such that:
-
-
(i.e.,
is a normal subgroup of
) for every ordinal
.
- If
is a limit ordinal, then
.
and such that there is some ordinal such that
.
In terms of the descendant closure operator
The subgroup property of being an descendant subgroup is obtained by applying the descendant closure operator to the subgroup property of being normal.
Relation with other properties
Stronger properties
Weaker properties
Related properties
Opposites
Facts
Descendant-contranormal factorization
This result states that given any subgroup of
, there is a unique subgroup
containing
such that
is contranormal in
and
is descendant in
.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | descendance is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
trim subgroup property | Yes | Every group is descendant in itself, and the trivial subgroup is descendant in any group. | |
intermediate subgroup condition | Yes | descendance satisfies intermediate subgroup condition | If ![]() ![]() ![]() ![]() ![]() |
strongly intersection-closed subgroup property | Yes | descendance is strongly intersection-closed | If ![]() ![]() ![]() |
image condition | No | descendance does not satisfy image condition | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() |