Semiabelian group

Semiabelian groups are a class of groups first introduced by and named by. It appears in Galois theory, in the study of the inverse Galois problem or the embedding problem which is a generalization of the former.

Definition

The family of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:
The class of finite groups G with a regular realizations over is closed under taking semidirect products with abelian kernels, and it is also closed under quotients. The class is the smallest class of finite groups that have both of these closure properties as mentioned above.

Example

Abelian groups, dihedral groups, and all -groups of order less than are semiabelian.
The following are equivalent for a non-trivial finite group G :
: G is semiabelian.
: G possess an abelian and a some proper semiabelian subgroup U with.
Finite semiabelian groups possess G-realizations over function fields in one variable for any field and therefore are Galois groups over every Hilbertian field.