Baer norm: Difference between revisions
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{{subgroup-defining function}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
The '''Baer norm''' of a [[group]] is the | The '''Baer norm''' of a [[group]] is defined in the following equivalent ways: | ||
* It is the intersection of [[normalizer]]s of all its [[subgroup]]s | |||
* It is the intersection of normalizers of all [[cyclic group|cyclic]] subgroups. | |||
* It is the set of those elements of the group for which the corresponding conjugation is a [[power automorphism]]. | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
The ''Baer norm''' of a [[group]] <math>G</math> is defined as the intersection, over all subgroups <math>H</math> of <math>G</math> of the groups <math>N_G(H)</math>. | The ''Baer norm''' of a [[group]] <math>G</math> is defined as the intersection, over all subgroups <math>H</math> of <math>G</math> of the groups <math>N_G(H)</math>. | ||
{{obtainedbyapplyingthe|intersect-all operator|normalizer subgroup}} | |||
==Property theory== | ==Property theory== | ||
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The Baer norm of a group contains the [[center]] of the group. Conjugation by any element in the Baer norm is a power automorphism, hence it commutes with all [[inner automorphism]]s. Thus, the Baer norm is contained in the second member of the [[upper central series]]. | The Baer norm of a group contains the [[center]] of the group. Conjugation by any element in the Baer norm is a power automorphism, hence it commutes with all [[inner automorphism]]s. Thus, the Baer norm is contained in the second member of the [[upper central series]]. | ||
Revision as of 09:23, 26 March 2007
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
Definition
Symbol-free definition
The Baer norm of a group is defined in the following equivalent ways:
- It is the intersection of normalizers of all its subgroups
- It is the intersection of normalizers of all cyclic subgroups.
- It is the set of those elements of the group for which the corresponding conjugation is a power automorphism.
Definition with symbols
The Baer norm' of a group is defined as the intersection, over all subgroups of of the groups .
In terms of the intersect-all operator
This property is obtained by applying the intersect-all operator to the property: normalizer subgroup
View other properties obtained by applying the intersect-all operator
Property theory
Reverse monotonicity
The Baer norm subgroup-defining function is weakly reverse monotone, that is, if is a subgroup of containing the Baer norm of , then the Baer norm of contains the Baer norm of .
Idempotence and iteration
The Baer norm of a group equals its own Baer norm. A group equals its own Baer norm if and only if it is a Hamiltonian group, that is, every subgroup in it is normal.
Quotient-idempotence and quotient-iteration
The quotient function corresponding to the Baer norm is not transitive.
Relation with other subgroup-defining functions
The Baer norm of a group contains the center of the group. Conjugation by any element in the Baer norm is a power automorphism, hence it commutes with all inner automorphisms. Thus, the Baer norm is contained in the second member of the upper central series.