Contrasaturated 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 inside $G$ if for any natural number $n$ and any $g\in G_n$, the following are true:

• There exists a natural number $n_1$ and a $g_1\in G_{n_1}$ such that ${\mathrm{\Phi }}_{n,n_1}(g,g_1)\in H_{n+n_1}$
• There exists a natural number $n_2$ and a $g_2\in G_{n_2}$ such that ${\mathrm{\Phi }}_{n_2,n}(g_2,g)\in H_{n_2+n}$

Another way of saying this is that the given sub-IAPS is not contained in any proper saturated sub-IAPS.

Examples

Contrasaturation via cancellation

• In the IAPS of symmetric groups, the sub-IAPS of alternating groups is contrasaturated: any element of the symmetric group $S_n$ can be mapped to an element inside $A_{n+2}$ (using the transposition for padding if it is outside $A_n$). In other words, we can pick $n_1=n_2=2$.
• In the GL IAPS, the sub-IAPS of special linear groups is contrasaturated: any element of $GL(n,R)$ lives inside $SL(n+1,R)$ (using the inverse of the determinant for padding). In other words, we can pick $n_1=n_2=1$.

Contrasaturation via expansion or absorption

• $GL(n,{\mathbb{F}}_{p^{2^n}})$ inside $GL(n,{\mathbb{F}}_{p^{2^{n^2}}})$ for prime $p$. This is contrasaturated because the field size of the sub-IAPS eventually grows to match the field size of the whole IAPS, and hence "absorbs" all elements of the whole IAPS. Importantly, any example of this flavor is not a padding-saturated sub-IAPS. The rapid growth in this case also prevents it from being a finitely displaced sub-IAPS.

Relation with other properties

Stronger properties

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

Incomparable properties

Property	Meaning	Proof of non-implication	Proof of non-implication (reverse)	Intermediate notions
finitely displaced sub-IAPS	injective interleaving but it need not be via a padding map	$GL(n,{\mathbb{F}}_{p^{2^n}})$ inside $GL(n,{\mathbb{F}}_{p^{2^{n^2}}})$ for prime $p$ (the intuition is that the IAPS grows so fast that there is no uniform displacement $n$ for which the sub-IAPS order at $m+n$ is at least the IAPS order at $m$ for all $m$, ruling out finite displacement by cardinality considerations alone.	general linear IAPS inside general affine IAPS (finitely displaced but not contrasaturated); see also power sub-APS of power APS may be finitely displaced, which immediately generates powerful counterexamples