Baer norm
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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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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
Group properties
The Baer norm is a Dedekind group, i.e., it is a group in which every subgroup is normal. Conversely, every Dedekind group equals its own Baer norm.
Further information: Baer norm is Dedekind
Subgroup properties
Properties satisfied
Property | Meaning | Proof of satisfaction |
---|---|---|
Normal subgroup | ||
Hereditarily permutable subgroup | every subgroup is a permutable subgroup of the whole group | Baer norm is hereditarily permutable |
Hereditarily 2-subnormal subgroup | every subgroup is a 2-subnormal subgroup of the whole group | Baer norm is hereditarily 2-subnormal |
Characteristic subgroup | invariant under all automorphisms | Baer norm is characteristic |
Strictly characteristic subgroup | invariant under all surjective endomorphisms | Baer norm is strictly characteristic |
Properties not satisfied
Property | Meaning | Proof of dissatisfaction |
---|---|---|
Fully invariant subgroup | invariant under all endomorphisms | Baer norm not is fully invariant |
Hereditarily normal subgroup | every subgroup is normal in the whole group | Baer norm not is hereditarily normal |
Examples
Dedekind groups
A Dedekind group is a group in which every subgroup is normal, or equivalently, a group that equals its own Baer norm. The finite Dedekind groups are precisely the following:
- finite abelian groups
- finite nilpotent groups whose 2-Sylow subgroup is a product of the quaternion group of order eight and an elementary abelian group, and all other Sylow subgroups are abelian.
The smallest examples of Dedekind non-abelian groups are quaternion group and direct product of Q8 and Z2.
Examples in groups of prime power order
Here are some examples where the Baer norm is a proper subgroup:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
Center of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z2 | Klein four-group |
Examples in other groups
Here are some examples in non-nilpotent groups:
Relation with other subgroup-defining functions
Smaller subgroup-defining functions
Subgroup-defining function | Meaning | Proof of containment | Proof of strictness |
---|---|---|---|
Center | Elements that commute with every element | Baer norm contains center | Center not contains Baer norm |
Larger subgroup-defining functions
Subgroup-defining function | Meaning | Proof of containment | Proof of strictness |
---|---|---|---|
Wielandt subgroup | intersection of normalizers of subnormal subgroup | Wielandt subgroup contains Baer norm | Baer norm not contains Wielandt subgroup |
Second center | second member of upper central series | Second center contains Baer norm | Baer norm not contains second center |
Centralizer of derived subgroup | centralizer of derived subgroup (commutator subgroup) | Centralizer of derived subgroup contains Baer norm | Baer norm not contains centralizer of derived subgroup |
Related subgroup properties
- Subgroup contained in the Baer norm is a subgroup contained in the Baer norm.
- Normal subgroup contained in the Baer norm is a normal subgroup of the whole group contained in the Baer norm.
Subgroup-defining function properties
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 Dedekind 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.