Difference between revisions of "Ascendant subgroup"

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(Metaproperties)
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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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| [[satisfies metaproperty::transitive subgroup property]] || Yes || || 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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| [[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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{{intsubcondn}}
 
 
 
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.
 
 
 
{{intersection-closed}}
 
 
 
An arbitrary intersection of ascendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.
 

Revision as of 21:19, 20 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

Symbol-free definition

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 H of a group G is termed ascendant if we have a series H_\alpha 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})
  • If \alpha is a limit ordinal, then H_\alpha = \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

Weaker properties

Opposites

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes 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.