Group satisfying subnormal join property: Difference between revisions
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A group is said to satisfy the '''subnormal join property''' if it satisfies the following equivalent conditions: | A group is said to satisfy the '''subnormal join property''' if it satisfies the following equivalent conditions: | ||
# The [[fact about::join of subgroups|join]] (i.e., subgroup generated) of two [[fact about::subnormal subgroup]]s of the group is again subnormal. | |||
# The join of a finite collection of subnormal subgroups of the group is again subnormal. | |||
# The [[fact about::commutator of two subgroups|commutator]] of any two subnormal subgroups of the group is again subnormal. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::group satisfying generalized subnormal join property]] || an arbitrary join of subnormal subgroups is subnormal || || || {{intermediate notions short|group satisfying subnormal join property|group satisfying generalized subnormal join property}} | |||
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| [[Weaker than::nilpotent group]] || has finite [[nilpotency class]] || || || {{intermediate notions short|group satisfying subnormal join property|nilpotent group}} | |||
|- | |||
| [[Weaker than::group in which every subgroup is subnormal]] || every subgroup is a [[subnormal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|group in which every subgroup is subnormal}} | |||
|- | |||
| [[Weaker than::simple group]] || nontrivial; no proper nontrivial [[normal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|simple group}} | |||
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| [[Weaker than::T-group]] || every subnormal subgroup is normal || || || {{intermediate notions short|group satisfying subnormal join property|T-group}} | |||
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| [[Weaker than::group in which every subnormal subgroup is 2-subnormal]] || every [[subnormal subgroup]] is a [[2-subnormal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|group in which every subnormal subgroup is 2-subnormal}} | |||
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| [[Weaker than::group with nilpotent derived subgroup]] || the [[derived subgroup]] is a [[nilpotent group]] || [[nilpotent derived subgroup implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group with nilpotent derived subgroup}} | |||
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| [[Weaker than::supersolvable group]] || has a [[normal series]] where all successive quotients are [[cyclic group]]s || || || {{intermediate notions short|group satisfying subnormal join property|supersolvable group}} | |||
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| [[Weaker than::finite group]] || has finite [[order of a group|order]] || [[finite implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|finite group}} | |||
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| [[Weaker than::Noetherian group]] (also called '''slender group''') || every subgroup is [[finitely generated group|finitely generated]] || [[Noetherian implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|Noetherian group}} | |||
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| [[Weaker than::group satisfying ascending chain condition on subnormal subgroups]] || no infinite strictly ascending chain of [[subnormal subgroup]]s || [[ascending chain condition on subnormal subgroups implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group satisfying ascending chain condition on subnormal subgroups}} | |||
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| [[Weaker than::group whose derived subgroup satisfies ascending chain condition on subnormal subgroups]] || derived subgroup contains no infinite strictly ascending chain of subnormal subgroups || [[derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group whose derived subgroup satisfies ascending chain condition on subnormal subgroups}} | |||
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==References== | ==References== | ||
Latest revision as of 22:42, 16 February 2011
This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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View a list of other standard non-basic definitions
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition
A group is said to satisfy the subnormal join property if it satisfies the following equivalent conditions:
- The join (i.e., subgroup generated) of two Subnormal subgroup (?)s of the group is again subnormal.
- The join of a finite collection of subnormal subgroups of the group is again subnormal.
- The commutator of any two subnormal subgroups of the group is again subnormal.
Relation with other properties
Stronger properties
References
Textbook references
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 388 (definition in paragraph)