Relative-intersection-closed subgroup property

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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Definition

A subgroup property $p$ is termed relative-intersection-closed if it satisfies the following:

Suppose $(I,<)$ is a well-ordered set. Suppose $H_{i},i\in I$ is a collection of subgroups of a group $G$ , such that for any $i\in I$ , $H_{i}$ satisfies property $p$ in some subgroup of $G$ containing both $H_{i}$ and $\bigcap _{j<i}H_{j}$ . Then: