Semiabelian group explained

Semiabelian group should not be confused with semiabelian scheme.

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

l{S}

of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:

G\inl{S}

acts on a finite abelian group

A

, then

A\rtimesG\inl{S}

;

G\inl{S}

and

N\triangleleftG

is a normal subgroup, then

G/N\inl{S}

.

The class of finite groups G with a regular realizations over

Q

is closed under taking semidirect products with abelian kernels, and it is also closed under quotients. The class

l{S}

is the smallest class of finite groups that have both of these closure properties as mentioned above.

Example

64

are semiabelian.

(i) G is semiabelian.

(ii) G possess an abelian

A\triangleleftG

and a some proper semiabelian subgroup U with

G=AU

.

Therefore G is an epimorphism of a split group extension with abelian kernel.

k(t)

in one variable for any field

k

and therefore are Galois groups over every Hilbertian field.

See also

References

Bibliography

Further reading

External links

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