# Intermediately operator preserves join-closedness

From Groupprops

## Statement

Suppose is a subgroup property and is the Intermediately operator (?). Then:

- If is a Join-closed subgroup property (?) (respectively, Strongly join-closed subgroup property (?)), so is .
- If is a Finite-join-closed subgroup property (?) (respectively, Strongly finite-join-closed subgroup property (?)), so is .

## Related facts

### Applications

- Intermediate characteristicity is strongly join-closed
- Intermediate full invariance is strongly join-closed