Absolute center

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:

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

Larger subgroup-defining functions

 * Contained in::Center: The fixed-point subgroup of the inner automorphism group of $$G$$.

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: