Descendant subgroup: Difference between revisions

Latest revision as of 22:47, 5 July 2019

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]
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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 |

Definition

A subgroup $H$ of a group $G$ is termed descendant if we have subgroups $H_{α}$ of $G$ for every ordinal $α$ such that:

$H_{0} = G$
$H_{α + 1} \underline{◃} H_{α}$ (i.e., $H_{α + 1}$ is a normal subgroup of $H_{α}$ ) for every ordinal $α$ .
If $α$ is a limit ordinal, then $H_{α} = ⋂_{γ < α} H_{γ}$ .

and such that there is some ordinal $β$ such that $H_{β} = H$ .

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

Facts

Descendant-contranormal factorization

This result states that given any subgroup $H$ of $G$ , there is a unique subgroup $K$ containing $H$ such that $H$ is contranormal in $K$ and $K$ is descendant in $G$ .

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	descendance is transitive	If $H \leq K \leq G$ are groups such that $H$ is a descendant subgroup of $K$ and $K$ is a descendant subgroup of $G$ , then $H$ is a descendant subgroup of $G$ .
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 $H \leq K \leq G$ are groups such that $H$ is descendant in $G$ , then $H$ is descendant in $K$ .
strongly intersection-closed subgroup property	Yes	descendance is strongly intersection-closed	If $H_{i}, i \in I$ , are all descendant subgroups of $G$ , so is the intersection $⋂_{i \in I} H_{i}$ .
image condition	No	descendance does not satisfy image condition	It is possible to have groups $G$ and $K$ , a descendant subgroup $H$ of $G$ and a surjective homomorphism $φ : G \to K$ such that $φ (H)$ is not a descendant subgroup of $K$ .

@@ Line 1: / Line 1: @@
 {{subgroup property}}
+{{finitarily equivalent to|subnormal subgroup}}
 {{variationof|subnormality}}
 ==Definition==
-===Symbol-free definition===
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''descendant''' if we have subgroups <math>H_\alpha</math> of <math>G</math> for every ordinal <math>\alpha</math> such that:
-A [[subgroup]] of a [[group]] is said to be '''descendant''' if there is a descending series of subgroups indexed by ordinals, each normal in its predecessor, that starts at the whole group, and terminates at the given subgroup.
+* <math>H_0 = G</math>
+* <math>H_{\alpha + 1} \ \underline{\triangleleft} \ H_\alpha</math> (i.e., <math>H_{\alpha + 1}</math> is a [[normal subgroup]] of <math>H_\alpha</math>) for every ordinal <math>\alpha</math>.
-===Definition with symbols===
+* If <math>\alpha</math> is a limit ordinal, then <math>H_\alpha = \bigcap_{\gamma < \alpha} H_\gamma</math>.
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''descendant''' if we have a series <math>H_\alpha</math> for every ordinal <math>\alpha</math> such that:
-* <math>H_0 = G</math>
+and such that there is some ordinal <math>\beta</math> such that <math>H_\beta = H</math>.
-* <math>H_{\alpha + 1} \triangleleft H_\alpha</math> (viz <math>H_{\alpha + 1}</math> is a [[normal subgroup]] of <math>H_\alpha</math>)
-* There is some <math>\beta</math> such that <math>H_\beta = G</math> (note that beyond this point we must get all <math>H_\alpha = G</math>).
 ===In terms of the descendant closure operator===
@@ Line 39: / Line 35: @@
 ===Opposites===
-* [[Self-normalizing subgroup]]
+* [[Contranormal subgroup]]
-==Metaproperties==
+==Facts==
-{{transitive}}
+===Descendant-contranormal factorization===
-A descendant subgroup of a descendant subgroup is descendant. The proof relies on simply ''concatenating'' the two descending series.
+This result states that given any subgroup <math>H</math> of <math>G</math>, there is a unique subgroup <math>K</math> containing <math>H</math> such that <math>H</math> is [[contranormal subgroup|contranormal]] in <math>K</matH> and <math>K</math> is [[descendant subgroup|descendant]] in <math>G</math>.
-{{trim}}
+==Metaproperties==
-The trivial subgroup and the whole group are both [[normal subgroup]]s, hence they are also both descendant subgroups.
-{{intsubcondn}}
-Any descendant subgroup of a group is also descendant in every intermediate subgroup. The proof of this follows by intersecting every member of the descending series with the intermediate subgroup and observing that normality at each stage is preserved.
-{{intersection-closed}}
-An arbitrary intersection of descendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[descendance is transitive]]|| If <math>H \le K \le G</math> are groups such that <math>H</math> is a descendant subgroup of <math>K</math> and <math>K</math> is a descendant subgroup of <math>G</math>, then <math>H</math> is a descendant subgroup of <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || || Every group is descendant in itself, and the trivial subgroup is descendant in any group.
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[descendance satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is descendant in <math>G</math>, then <math>H</math> is descendant in <math>K</math>.
+|-
+| [[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[descendance is strongly intersection-closed]] || If <math>H_i, i \in I</math>, are all descendant subgroups of <math>G</math>, so is the intersection <math>\bigcap_{i \in I} H_i</math>.
+|-
+| [[dissatisfies metaproperty::image condition]] || No || [[descendance does not satisfy image condition]] || It is possible to have groups <math>G</math> and <math>K</math>, a descendant subgroup <math>H</math> of <math>G</math> and a surjective homomorphism <math>\varphi:G \to K</math> such that <math>\varphi(H)</math> is not a descendant subgroup of <math>K</math>.
+|}