Finitely displaced normal sub-IAPS

This article defines a group sub-IAPS property: a property that can be evaluated to true/false given an IAPS of groups and a sub-IAPS.

Definition

Let $(G, Φ)$ be an IAPS of groups and $H$ be a sub-IAPS. We say that $H$ is a finitely displaced normal sub-IAPS if $H$ is both finitely displaced and normal in $G$ .

In the IAPS of symmetric groups, the sub-IAPS of alternating groups is finitely displaced: we can use the homomorphism from $S_{n}$ to $A_{n + 2}$ that pads with the identity element if inside $A_{n}$ a and with the transposition outside $A_{n}$ .
In the GL IAPS, the sub-IAPS of special linear groups is finitely displaced: any element of $G L (n, R)$ lives inside $S L (n + 1, R)$ that pads with the inverse of the determinant.

In these examples, the finite displacement strategy does not involve concatenation.

Power sub-APS of power APS may be finitely displaced uses the fact that infinity can absorb anything finite to demonstrate finite displacement.

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finitely displaced sub-IAPS	remove the normality assumption	(obvious)	general linear IAPS inside general affine IAPS	\|FULL LIST, MORE INFO

Property	Meaning	Proof of non-implication	Proof of non-implication (reverse)	Intermediate notions
contrasaturated normal sub-IAPS	every element of the IAPS can be concatenated into the sub-IAPS	follows from the generic abelian example for power sub-APS of power APS may be finitely displaced (finitely displaced normal not implies contrasaturated)	constant-abelian-quotient not implies finitely displaced + eventually constant-abelian-quotient implies contrasaturated
eventually constant-abelian-quotient sub-IAPS	normal sub-IAPS whose quotient APS is eventually isomorphic to the constant APS for an abelian group	(via contrasaturated normal)	constant-abelian-quotient not implies finitely displaced