Absolute center

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

History

The concept appears to have first been systematically discussed in Hegarty's 1994 paper.

Definition

The absolute center of a group $G$, sometimes denoted $L(G)$ is defined as the fixed-point subgroup in $G$ under the action of the whole automorphism group $\operatorname {Aut} (G)$. In symbols, it is the subset:

$\{g\in G\mid \sigma (g)=g\ \forall \ \sigma \in \operatorname {Aut} (G)\}$

Relation with other subgroup-defining functions

Larger subgroup-defining functions

• Center: The fixed-point subgroup of the inner automorphism group of $G$.

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 $n^{th}$ 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}