# Strongly join-closed subgroup property

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

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Contents

## Definition

A subgroup property is termed **strongly join-closed** if for any (possibly empty, possibly finite and possibly infinite) collection of subgroups of a group such that each satisfies in , the join of subgroups also satisfies in .

By convention, if 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 |