P-simple IAPS

Symbol-free definition
An IAPS of groups is termed p-simple if it has no strongly proper nontrivial normal sub-IAPS.

Definition with symbols
An IAPS of groups $$(G,\Phi)$$ is termed p-simple if there is no sub-IAPS $$H$$ of $$G$$ satisfying all these conditions:


 * $$H_n \triangleleft G_n$$ for every $$n$$
 * There are infinitely many indices $$n$$ for which $$H_n$$ is properly contained in $$G_n$$
 * $$H_n$$ is nontrivial for at least some value of $$n$$

Stronger properties

 * Eventually simple IAPS

Weaker properties

 * i-simple IAPS