ACU-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Symbol-free definition

A subgroup property is termed ACU-closed (or closed under ascending chain unions) if given any ascending chain of subgroups, each of which has the property, the union of those subgroups also has the property. The ascending chain here is indexed by natural numbers.

A somewhat stronger notion is that of TACU-closed subgroup property, which also allows for transfinite ascending chains.

Definition with symbols

A subgroup property <math>p</math> is termed ACU-closed (or closed under ascending chain unions) if given an ascending chain of subgroups <math>H_1</math> &le; <math>H_2</math> &le; <math>H_3</math> ... in a group <math>G</math> such that each <math>H_i</math> satisfies property <math>p</math>, the union of all the <math>H_i</math>s also satisfies <math>p</math>.