Ascendant 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 ascendant if we have subgroups
of
for every ordinal
such that:
-
-
(viz
is a normal subgroup of
) for every ordinal
.
- If
is a limit ordinal, then
, i.e., it is the union of all preceding subgroups. Note that the union of any ascending chain of subgroups is indeed a sugbroup (in fact, more generally, directed union of subgroups is subgroup). We can also define
as
, i.e., it is the join of all preceding subgroups.
and such that there is some ordinal such that
.
In terms of the ascendant closure operator
The subgroup property of being an ascendant subgroup is obtained by applying the ascendant closure operator to the subgroup property of being normal.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
normal subgroup | we can get a series that reaches the group in one step. | |||
subnormal subgroup | we can get a series that reaches the group in finitely many steps. | (obvious) | ascendant not implies subnormal | |FULL LIST, MORE INFO |
hypernormalized subgroup | we can use the series where the subgroup for each successor ordinal is the normalizer in the whole group of the subgroup for the ordinal. | (obvious) | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | |
---|---|---|---|---|---|
serial subgroup |
Opposites
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | ascendance is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
trim subgroup property | Yes | Every group is ascendant in itself, and the trivial subgroup is ascendant in any group. | |
intermediate subgroup condition | Yes | ascendance satisfies intermediate subgroup condition | If ![]() ![]() ![]() ![]() ![]() |
strongly intersection-closed subgroup property | Yes | ascendance is strongly intersection-closed | If ![]() ![]() ![]() |
transfer condition | Yes | ascendance satisfies transfer condition | If ![]() ![]() ![]() ![]() ![]() |
image condition | Yes | ascendance satisfies image condition | Suppose ![]() ![]() ![]() ![]() ![]() |