Semiabelian group
Semiabelian groups is 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:
 If  acts on a finite abelian group , then ;
 If  and  is a normal subgroup, then . 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 the 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  :
(i) G is semiabelian. (ii) G posses an abelian  and a some proper semiabelian subgroup U with . Therefore G is an epimorphism of a split group extension with abelian kernel. Finite semiabelian groups possess G-realizations over function fields  in one variable for any field  and therefore are Galois groups over every Hilbertian field. See also 
Inverse Galois problem
Embedding problem
Galois extension

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Galois theory