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Strongly join-closed subgroup property

Definition

A subgroup property p is termed strongly join-closed if for any (possibly empty, possibly finite and possibly infinite) collection H_i, i \in I of subgroups of a group G such that each H_i satisfies p in G, the join of subgroups \langle H_i \rangle_{i \in I} also satisfies p in G.

By convention, if I is empty, the join is taken as the trivial subgroup. In particular, a subgroup property is strongly join-closed if and only if it is join-closed and trivially true (i.e., always satisfied by the trivial subgroup).

Relation with other metaproperties

Stronger metaproperties

Metaproperty name Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Strongly UL-join-closed subgroup property

Weaker metaproperties

Metaproperty name Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
join-closed subgroup property
finite-join-closed subgroup property
strongly finite-join-closed subgroup property