Contrasaturated 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 $(G,\mathrm{\Phi })$ be an IAPS of groups and $H$ a sub-IAPS of $G$. We say that $H$ is contrasaturated normal inside $G$ if $H$ is both contrasaturated and a normal.

Examples

Contrasaturation via cancellation

• In the GL IAPS, the sub-IAPS comprising special linear groups is a contrasaturated normal sub-IAPS. The quotient APS is the constant APS of the group of units of the ring.
• In the IAPS of symmetric groups, the sub-IAPS comprising alternatig groups is a contrasaturated normal sub-IAPS. The quotient APS is the trivial group at $n=1$ and after that is cyclic group:Z2, so it is eventually the constant APS of cyclic group:Z2.
• The example for constant-abelian-quotient not implies finitely displaced is a non-split example where we have cancellation, but without splitting, which means no finite displacement.

Contrasaturation via expansion or absorption

• In the IAPS $GL(n,{\mathbb{F}}_{p^{2^n}})$ for a prime number $p$, the sub-IAPS $SL(n,{\mathbb{F}}_{p^{2^n}})$ is contrasaturated and normal for the same reason as that for the SL IAPS in the GL IAPS. Unlike the two examples above, the quotient APS, whose members are the multiplicative groups of ${\mathbb{F}}_{p^{2^n}}$ with the concatenation maps being multiplication in the field (with natural field inclusion), is not constant or bounded.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
eventually constant-abelian-quotient sub-IAPS	normal sub-IAPS whose quotient APS is eventually isomorphic to the constant APS of an abelian group	eventually constant-abelian-quotient implies contrasaturated	$SL(n,{\mathbb{F}}_{p^{2^n}})$ inside $GL(n,{\mathbb{F}}_{p^{2^n}})$	|FULL LIST, MORE INFO
constant-abelian-quotient sub-IAPS	normal sub-IAPS whose quotient APS is the constant APS of an abelian group	(via eventually constant-abelian-quotient)	(via eventually constant-abelian-quotient)	Eventually constant-abelian-quotient sub-IAPS|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
contrasaturated sub-IAPS	drop the normality assumption	(obvious)	$GL(n,{\mathbb{F}}_{p^{2^n}})$ inside $GL(n,{\mathbb{F}}_{p^{2^{n^2}}})$ for prime $p$	|FULL LIST, MORE INFO

Incomparable properties

Property	Meaning	Proof of non-implication	Proof of non-implication (reverse)	Intermediate notions
finitely displaced normal sub-IAPS	IAPS embeds in sub-IAPS member-wise	constant-abelian-quotient not implies finitely displaced + eventually constant-abelian-quotient implies contrasaturated	follows from the generic abelian example for power sub-APS of power APS may be finitely displaced (finitely displaced normal not implies contrasaturated)