Difference between revisions of "Ascendant subgroup"

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(Metaproperties)
 
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==Definition==
 
==Definition==
  
===Symbol-free definition===
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''ascendant''' 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 '''ascendant''' if there is an ascending series of subgroups indexed by ordinals, each normal in the next, that starts at the given subgroup, and terminates at the whole group.
 
 
 
===Definition with symbols===
 
 
 
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''ascendant''' if we have a series <math>H_\alpha</math> for every ordinal <math>\alpha</math> such that:
 
  
 
* <math>H_0 = H</math>
 
* <math>H_0 = H</math>
* <math>H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}</math> (viz <math>H_\alpha</math> is a [[normal subgroup]] of <math>H_{\alpha + 1}</math>)
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* <math>H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}</math> (viz <math>H_\alpha</math> is a [[normal subgroup]] of <math>H_{\alpha + 1}</math>) for every ordinal <math>\alpha</math>.
* If <math>\alpha</math> is a limit ordinal, then <math>H_\alpha = \langle H_\gamma \rangle_{\gamma < \alpha}</math>, i.e., it is the [[join of subgroups|join]] of all preceding subgroups.
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* If <math>\alpha</math> is a limit ordinal, then <math>H_\alpha = \bigcup_{\gamma < \alpha} H_\gamma</math>, 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 <math>H_{\alpha}</math>as <math>\langle H_\gamma \rangle_{\gamma < \alpha}</math>, i.e., it is the [[join of subgroups|join]] of all preceding subgroups.
  
 
and such that there is some ordinal <math>\beta</math> such that <math>H_\beta = G</math>.
 
and such that there is some ordinal <math>\beta</math> such that <math>H_\beta = G</math>.
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===Stronger properties===
 
===Stronger properties===
  
* [[Normal subgroup]]
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{| class="sortable" border="1"
* [[Subnormal subgroup]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Hypernormalized subgroup]]
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|-
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| [[Weaker than::normal subgroup]] || we can get a series that reaches the group in one step. || || ||
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|-
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| [[Weaker than::subnormal subgroup]] || we can get a series that reaches the group in finitely many steps. || (obvious) || [[ascendant not implies subnormal]] || {{intermediate notions short|ascendant subgroup|subnormal subgroup}}
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|-
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| [[Weaker than::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) || || {{intermediate notions short|ascendant subgroup|hypernormalized subgroup}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Serial subgroup]]
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{| class="sortable" border="1"
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Stronger than::serial subgroup]] || || || || ||
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|}
  
 
===Opposites===
 
===Opposites===
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==Metaproperties==
 
==Metaproperties==
  
{{transitive}}
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{| class="sortable" border="1"
 
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
An ascendant subgroup of an ascendant subgroup is ascendant. The proof relies on simply ''concatenating'' the two ascending series.
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|-
 
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| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[ascendance is transitive]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is an ascendant subgroup of <math>K</math> and <math>K</math> is an ascendant subgroup of <math>G</math>, then <math>H</math> is an ascendant subgroup of <math>G</math>.
{{trim}}
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|-
 
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| [[satisfies metaproperty::trim subgroup property]] || Yes || || Every group is ascendant in itself, and the trivial subgroup is ascendant in any group.
Thetrivial subgroup and the whole group are both [[normal subgroup]]s, hence they are also both ascendant subgroups.
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|-
 
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| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[ascendance satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is an ascendant subgroup of <math>G</math>, then <math>H</math> is an ascendant subgroup of <math>K</math>.
{{intsubcondn}}
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|-
 
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| [[satisfies metaproprty::strongly intersection-closed subgroup property]] || Yes || [[ascendance is strongly intersection-closed]] || If <math>H_i, i \in I</math>, are all ascendant subgroups of a group <math>G</math>, so is <math>\bigcap_{i \in I} H_i</math>.
Any ascendant subgroup of a group is also ascendant in every intermediate subgroup. The proof of this follows by intersecting every member of the ascending series with the intermediate subgroup and observing that normality at each stage is preserved.
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|-
 
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| [[satisfies metaproperty::transfer condition]] || Yes || [[ascendance satisfies transfer condition]] || If <math>H,K \le G</math> are subgroups and <math>H</math> is ascendant in <math>G</math>, then <math>H \cap K</math> is ascendant in <math>K</math>.
{{intersection-closed}}
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|-
 
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| [[satisfies metaproperty::image condition]] || Yes || [[ascendance satisfies image condition]] || Suppose <math>H</math> is an ascendant subgroup of a group <math>G</math>, and <math>\varphi:G \to K</math> is a surjective homomorphism of groups. Then, <math>\varphi(H)</math> is an ascendant subgroup of <math>K</math>.
An arbitrary intersection of ascendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.
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|}

Latest revision as of 17:47, 21 December 2012

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 |

Definition

A subgroup H of a group G is termed ascendant if we have subgroups H_\alpha of G for every ordinal \alpha such that:

  • H_0 = H
  • H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1} (viz H_\alpha is a normal subgroup of H_{\alpha + 1}) for every ordinal \alpha.
  • If \alpha is a limit ordinal, then H_\alpha = \bigcup_{\gamma < \alpha} H_\gamma, 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 H_{\alpha}as \langle H_\gamma \rangle_{\gamma < \alpha}, i.e., it is the join of all preceding subgroups.

and such that there is some ordinal \beta such that H_\beta = G.

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 H \le K \le G are groups such that H is an ascendant subgroup of K and K is an ascendant subgroup of G, then H is an ascendant subgroup of G.
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 H \le K \le G are groups such that H is an ascendant subgroup of G, then H is an ascendant subgroup of K.
strongly intersection-closed subgroup property Yes ascendance is strongly intersection-closed If H_i, i \in I, are all ascendant subgroups of a group G, so is \bigcap_{i \in I} H_i.
transfer condition Yes ascendance satisfies transfer condition If H,K \le G are subgroups and H is ascendant in G, then H \cap K is ascendant in K.
image condition Yes ascendance satisfies image condition Suppose H is an ascendant subgroup of a group G, and \varphi:G \to K is a surjective homomorphism of groups. Then, \varphi(H) is an ascendant subgroup of K.