Semiabelian group

Short description: Added a basic definition in group theory and algebra

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

[math]\displaystyle{ G_0 = \{1\}, G_1, \dots , G_n = G }[/math]

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

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

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

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

  (i) G is semiabelian.

  (ii) G posses an abelian [math]\displaystyle{ A\triangleleft G }[/math] and a some proper semiabelian subgroup U with [math]\displaystyle{ G = AU }[/math].

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

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

See also

• Inverse Galois problem
• Embedding problem
• Galois extension

References

Bibliography

• Blum-Smith, Benjamin (2014). "Semiabelian Groups and the Inverse Galois Problem". Courant Institute of Mathematical Sciences. https://math.nyu.edu/dynamic/calendars/seminars/algebraic-geometry-seminar/790/. 
• 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. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002236435. 
• 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. 
• 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. doi:10.1016/j.jalgebra.2021.12.026. 
• Matzat, Bernd Heinrich (1987). "Einbettungsprobleme Über Hilbertkörpern". Konstruktive Galoistheorie. Lecture Notes in Mathematics. 1284. pp. 215–268. doi:10.1007/BFb0098329. ISBN 978-3-540-18444-7. https://books.google.com/books?id=awB8CwAAQBAJ&pg=PA215. 
• 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. https://books.google.com/books?id=kovnCAAAQBAJ&pg=PA300. 
• 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. https://books.google.com/books?id=Y8caCAAAQBAJ&pg=PA33. 
• Neftin, Danny (2011). "On semiabelian p-groups". Journal of Algebra 344: 60–69. 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". Algebra & Number Theory 12 (10): 2387–2401. doi:10.2140/ant.2018.12.2387. https://projecteuclid.org/journals/algebra-and-number-theory/volume-12/issue-10/Realizing-2-groups-as-Galois-groups-following-Shafarevich-and-Serre/10.2140/ant.2018.12.2387.pdf. 
• 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. http://dx.doi.org/10.1016/0021-8693(84)90228-x. 

Further reading

• Matzat, B. Heinrich (1991). "Der Kenntnisstand in der konstruktiven Galoisschen Theorie". Representation Theory of Finite Groups and Finite-Dimensional Algebras. pp. 65–98. doi:10.1007/978-3-0348-8658-1_4. ISBN 978-3-0348-9720-4. https://books.google.com/books?id=LyXyBwAAQBAJ&pg=PA65. 
• 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". https://people.maths.bris.ac.uk/~matyd/InvGal/.