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

${\displaystyle G_{0}=\{1\},G_{1},\dots ,G_{n}=G}$

such that ${\displaystyle G_{i}}$ is a homomorphic image of a semidirect product ${\displaystyle A_{i}\rtimes G_{i-1}}$ with a finite abelian group ${\displaystyle A_{i}}$ (${\displaystyle i=1,\dots ,n}$.).

The family ${\displaystyle {\mathcal {S}}}$ of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:

• If ${\displaystyle G\in {\mathcal {S}}}$ acts on a finite abelian group ${\displaystyle A}$, then ${\displaystyle A\rtimes G\in {\mathcal {S}}}$;
• If ${\displaystyle G\in {\mathcal {S}}}$ and ${\displaystyle N\triangleleft G}$ is a normal subgroup, then ${\displaystyle G/N\in {\mathcal {S}}}$.
  

The class of finite groups G with a regular realizations over ${\displaystyle \mathbb {Q} }$ is closed under taking semidirect products with abelian kernels, and it is also closed under quotients. The class ${\displaystyle {\mathcal {S}}}$ 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 ${\displaystyle 64}$ are semiabelian.
• The following are equivalent for a non-trivial finite group G (Dentzer 1995, Theorm 2.3.) :

  (i) G is semiabelian.

  (ii) G possess an abelian ${\displaystyle A\triangleleft G}$ and a some proper semiabelian subgroup U with ${\displaystyle G=AU}$.

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

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

See also

• Inverse Galois problem
• Embedding problem
• Galois extension

References

Citations

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

• Blum-Smith, Benjamin (2014). "Semiabelian Groups and the Inverse Galois Problem". Courant Institute of Mathematical Sciences.
• 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. S2CID 119155359.
• Dentzer, Ralf (1995). "On geometric embedding problems and semiabelian groups". Manuscripta Mathematica. 86: 199–216. doi:10.1007/BF02567989. S2CID 122932323. 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. arXiv:0912.1964. doi:10.2140/ant.2010.4.1077. S2CID 73636129. Zbl 1221.11218.
• Kisilevsky, Hershy; Sonn, Jack (2010). "On the minimal ramification problem for ℓ-groups". Compositio Mathematica. 146 (3): 599–606. arXiv:0811.2978. doi:10.1112/S0010437X10004719. S2CID 16101476.
• 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. S2CID 245424796.
• Matzat, Bernd Heinrich (1987). "Einbettungsprobleme Über Hilbertkörpern". Konstruktive Galoistheorie. Lecture Notes in Mathematics (in German). Vol. 1284. pp. 215–268. doi:10.1007/BFb0098329. ISBN 978-3-540-18444-7.
• 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.
• Matzat, B. H. (1995). "Parametric solutions of embedding problems". Recent Developments in the Inverse Galois Problem. Contemporary Mathematics. Vol. 186. pp. 33–50. doi:10.1090/conm/186/02174. ISBN 9780821802991.
• Neftin, Danny (2011). "On semiabelian p-groups". Journal of Algebra. 344: 60–69. arXiv:0908.1472. doi:10.1016/j.jalgebra.2011.07.016. S2CID 16647073.
  • Neftin, Danny (2009). "On semiabelian p-groups". arXiv:0908.1472v2 .
• Stoll, Michael (1995). "Construction of semiabelian Galois extensions". Glasgow Mathematical Journal. 37: 99–104. doi:10.1017/S0017089500030433. S2CID 122194283.
• 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. S2CID 126693959.
• 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

• Matzat, B. Heinrich (1991). "Der Kenntnisstand in der konstruktiven Galoisschen Theorie". Representation Theory of Finite Groups and Finite-Dimensional Algebras (in German). pp. 65–98. doi:10.1007/978-3-0348-8658-1_4. ISBN 978-3-0348-9720-4.
• 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 links

• "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.

• Galois theory