Galois-realizable group

Definition with symbols
A group $$G$$ is said to be Galois-realizable over a field $$K$$ if there exists a normal field extension $$L$$ of $$K$$ such that $$G$$ is the Galois group of $$L/K$$.

If no field is specified, we assume that the field is $$K = \mathbb{Q}$$, the field of rational numbers.

The inverse Galois problem essentially asks for which finite groups can be expressed as the Galois group of a field extension of $$\mathbb{Q}$$.

Definition in terms of the universal Galois group
A group $$G$$ is Galois-realizable over a field $$K$$ if it is the quotient of the universal Galois group of $$K$$ by a normal subgroup which is closed under the naturally given Krull topology.

Weaker properties
The following properties are known to be weaker than the property of being Galois-realizable over $$\mathbb{Q}$$:


 * Solvable group