Absolute center
From Groupprops
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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Contents
History
The concept appears to have first been systematically discussed in Hegarty's 1994 paper.
Definition
The absolute center of a group , sometimes denoted is defined as the fixed-point subgroup in under the action of the whole automorphism group . In symbols, it is the subset:
Relation with other subgroup-defining functions
Larger subgroup-defining functions
- Center: The fixed-point subgroup of the inner automorphism group of .
Subgroup properties
Properties satisfied
Property | Meaning | Proof of satisfaction |
---|---|---|
central subgroup | contained in the center | by definition |
central factor | product with centralizer is whole group | follows from being a central subgroup |
hereditarily normal subgroup | every subgroup of it is normal | follows from being central |
characteristic subgroup | invariant under all automorphisms | every element in the subgroup is invariant under all automorphisms, so the whole group is |
fixed-point subgroup of a subgroup of the automorphism group | there is a subgroup of the automorphism group for which this is precisely the set of fixed points | in this case, the subgroup in question is the whole automorphism group |
local powering-invariant subgroup | unique root (in the whole group) of any element in the subgroup must be in the subgroup | follows from being a fixed-point subgroup of a subgroup of the automorphism group, along with the fact that fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant |
powering-invariant subgroup | powered for all primes that power the group | follows from being local powering-invariant |
quotient-powering-invariant subgroup | the quotient group is powered over all primes that the whole group is powered over. | followed from powering-invariant and central implies quotient-powering-invariant |
Properties not satisfied
In general, any example that shows that the center does not have a given property, and where the center is cyclic group:Z2, can be used to show that the absolute center also does not have the property. Below is a partial (to be expanded) list:
Property | Meaning | Proof of dissatisfaction |
---|---|---|
fully invariant subgroup | invariant under all endomorphisms | example for center not is fully invariant where the center is cyclic group:Z2 works, because in this case the absolute center must equal the center. |
References
- The Absolute Center of a Group by P. Hegarty, Journal of Algebra, ISSN 00218693, Volume 169,Number 3, Page 929 - 935(November 1994): ^{Official copy (gated)}^{More info}