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
"Definition:  A finite group G is called semiabelian if and only if there exists a sequence

$G_0 = \{1\}, G_1, \dots, G_n = G$ such that $G_i$ is a homomorphic image of a semidirect product $A_i\rtimes G_{i-1}$ with a finite abelian group $A_{i}$ ($i = 1, \dots, n$.)."

The family $$\mathcal{S}$$ of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:
 * If $$G \in \mathcal{S}$$ acts on a finite abelian group $$A$$, then $$A\rtimes G\in \mathcal{S}$$;
 * If $$G\in \mathcal{S}$$ and $$N\triangleleft G$$ is a normal subgroup, then $$G/N\in \mathcal{S}$$.

The class of finite groups G with a regular realizations over $$\mathbb{Q}$$ is closed under taking semidirect products with abelian kernels, and it is also closed under quotients. The class $$\mathcal{S}$$ is the smallest class of finite groups that have both of these closure properties as mentioned above.

Example

 * Abelian groups, dihedral groups, and all $p$-groups of order less than $$64$$ are semiabelian.
 * The following are equivalent for a non-trivial finite group G :  
 * (i) G is semiabelian.
 * (ii) G possess an abelian $$A\triangleleft G$$ and a some proper semiabelian subgroup U with $$G = AU$$.
 * Therefore G is an epimorphism of a split group extension with abelian kernel.


 * Finite semiabelian groups possess G-realizations over function fields $$k(t)$$ in one variable for any field $$k$$ and therefore are Galois groups over every Hilbertian field.