Normal sub-APS

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This term is related to: APS theory
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ANALOGY: This is an analogue in APS of a property encountered in group. Specifically, it is a sub-APS property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in APSs of subgroup properties (OR, View as a tabulated list)

Definition

Let $(G,\mathrm{\Phi })$ be an APS of groups and $H$ be a sub-APS of $G$. We say that $H$ is normal in $G$ if the following equivalent conditions hold:

• For every $n$, $H_n$ is a normal subgroup of $G_n$
• There exists an APS of groups $(K,\mathrm{\Psi })$ and an APS homomorphism from $(G,\mathrm{\Phi })$ to $(K,\mathrm{\Psi })$ such that the kernel of the map at each $n$ is $H_n$

Relation with other properties

Stronger properties

• Saturated normal sub-APS