On compactness of logics that can express properties of symmetry or connectivity

A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. We also give an example of a logic that extends first-order logic, has the compactness property and can express the property "the cardinality of the automorphism group is at most $2^{\aleph_0}$".