Semiabelian group (Galois theory)

Not to be confused with semiabelian scheme.

In Galois theory, the Semibelian group is a groups associated with the inverse Galois problem or the embedding problem which is a generalization of the former, introduce by Thompson (1984) (called "semiabelian" by Matzat (1987)).

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 $A\rtimes G$ 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 $G/N\in {\mathcal {S}}$.
  

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

Example

A non-trivial finite group G the followsing the are equivalent (Dentzer 1995, Theorm 2.3.) :

• (i) G is semiabelian.

  (ii) G posses 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-realization over any field and therefore are Galois groups over every Hilbertian field.

See also

• Inverse Galois problem
• Embedding problem
• Minimal ramification problem (Inverse Galois problem)
• p-group
• Galois extension

References

• De Witt, Meghan (2014). "Minimal ramification and the inverse Galois problem over the rational function field Fp(t)". Journal of Number Theory. 143: 62–81. doi:10.1016/j.jnt.2014.03.017.
• Dentzer, Ralf (1995). "On geometric embedding problems and semiabelian groups". Manuscripta Mathematica. 86: 199–216. doi:10.1007/BF02567989. Zbl 0836.12002.
• Kisilevsky, Hershy; Neftin, Danny; Sonn, Jack (2010). "On the minimal ramification problem for semiabelian groups". Algebra & Number Theory. 4 (8): 1077–1090. doi:10.2140/ant.2010.4.1077. Zbl 1221.11218.
• Kisilevsky, Hershy; Sonn, Jack (2010). "On the minimal ramification problem for ℓ-groups". Compositio Mathematica. 146 (3): 599–606. doi:10.1112/S0010437X10004719.
• Legrand, François (2022). "On finite embedding problems with abelian kernels". Journal of Algebra. 595: 633–659. arXiv:2112.12170. doi:10.1016/j.jalgebra.2021.12.026.
• Matzat, Bernd Heinrich (1987). "Einbettungsprobleme Über Hilbertkörpern". Konstruktive Galoistheorie. Lecture Notes in Mathematics (in Deutsch). 1284. pp. 215–268. doi:10.1007/BFb0098329. ISBN 978-3-540-18444-7. Search this book on
• Matzat, B. Heinrich (1991). "Der Kenntnisstand in der konstruktiven Galoisschen Theorie". Representation Theory of Finite Groups and Finite-Dimensional Algebras (in Deutsch). pp. 65–98. doi:10.1007/978-3-0348-8658-1_4. ISBN 978-3-0348-9720-4. Search this book on
• Malle, Gunter; Matzat, B. Heinrich (1999). "Embedding Problems". Inverse Galois Theory. Springer Monographs in Mathematics. pp. 263–360. doi:10.1007/978-3-662-12123-8_4. ISBN 978-3-662-12123-8. Search this book on
• Matzat, B. H. (1995). "Parametric solutions of embedding problems". Recent Developments in the Inverse Galois Problem. Contemporary Mathematics. 186. pp. 33–50. doi:10.1090/conm/186/02174. ISBN 9780821802991. Search this book on
• Neftin, Danny (2011). "On semiabelian p-groups". Journal of Algebra. 344: 60–69. arXiv:0908.1472. doi:10.1016/j.jalgebra.2011.07.016.
  • Neftin, Danny (2009). "On semiabelian p-groups". arXiv:0908.1472v2 [math.GR].
• Stoll, Michael (1995). "Construction of semiabelian Galois extensions". Glasgow Mathematical Journal. 37: 99–104. doi:10.1017/S0017089500030433.
• Schmid, Peter (2018). "Realizing 2-groups as Galois groups following Shafarevich and Serre" (PDF). Algebra & Number Theory. 12 (10): 2387–2401. doi:10.2140/ant.2018.12.2387.
• Thompson, John G (1984). "Some finite groups which appear as gal L/K, where K ⊆ Q(μn)". Journal of Algebra. 89 (2): 437–499. doi:10.1016/0021-8693(84)90228-x. ISSN 0021-8693.

Further reading

• Saltman, David J. (1982). "Generic Galois extensions and problems in field theory". Advances in Mathematics. 43 (3): 250–283. doi:10.1016/0001-8708(82)90036-6.

External link

• Blum-Smith, Benjamin (2014). "Semiabelian Groups and the Inverse Galois Problem". Courant Institute of Mathematical Sciences.
• "Inverse Galois Problem (Lecture 3) in PCMI 2021 Graduate Summer School Program - Number Theory Informed by Computation - July 26-30, 2021". Archived from the original on 2023-02-16.

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