Semiabelian groups is a class of groups first introduced by Thompson (1984) and named by Matzat (1987).[1] 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

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Definition:[2][3][4][5] A finite group G is called semiabelian if and only if there exists a sequence

 

such that   is a homomorphic image of a semidirect product   with a finite abelian group   ( .).

The family   of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:[6][7]

  • 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 these closure properties as mentioned above.[8][9]

Example

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  • Abelian groups, dihedral groups, and all p-groups of order less than   are semiabelian. [10]
  • The following are equivalent for a non-trivial finite group G (Dentzer 1995, Theorm 2.3.) :[11][12]
    (i) G is semiabelian.
    (ii) G possess an abelian   and a some proper semiabelian subgroup U with  .
Therefore G is an epimorphism of a split group extension with abelian kernel.[13]

See also

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References

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Citations

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  1. ^ (Stoll 1995)
  2. ^ (Dentzer 1995, Definition 2.1)
  3. ^ (Kisilevsky, Neftin & Sonn 2010)
  4. ^ (Kisilevsky & Sonn 2010)
  5. ^ (De Witt 2014)
  6. ^ (Thompson 1984)
  7. ^ (Neftin 2009, Definition 1.1.)
  8. ^ (Blum-Smith 2014)
  9. ^ (Legrand 2022)
  10. ^ Dentzer 1995.
  11. ^ (Matzat 1995, §6. Split extensions with Abelian kernel, Proposition 4)
  12. ^ (Neftin 2011)
  13. ^ (Schmid 2018)
  14. ^ (Malle & Matzat 1999, p. 33)
  15. ^ (Matzat 1995, p. 41)
  16. ^ (Malle & Matzat 1999, p. 300)

Bibliography

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Further reading

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