Semiabelian group

Template:Short description Template:Distinguish Semiabelian groups are a class of groups first introduced by Template:Harvtxt and named by Template:Harvtxt. 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 $𝒮$ of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:

• If $G\in 𝒮$ acts on a finite abelian group $A$, then $A⋊G\in 𝒮$;
• If $G\in 𝒮$ and $N◃G$ is a normal subgroup, then $G/N\in 𝒮$.
  

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 $𝒮$ 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-group|Template:Mvar-group]]s of order less than $64$ are semiabelian.Template:Sfn
• The following are equivalent for a non-trivial finite group G Template:Harv:

  (i) G is semiabelian.

  (ii) G possess an abelian $A◃G$ 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.

References

Bibliography

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Further reading

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External links

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