Embedding Obstructions for the Dihedral, Semidihedral, and Quaternion 2-Groups

✍ Scribed by Ivo M Michailov

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 134 KB
• Volume: 245
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8926

✦ Synopsis

For each of the dihedral, semidihedral, and quaternion 2-groups, we represent the obstructions to certain Brauer problems as tensor products of quaternion algebras. Then we reduce various embedding problems with cyclic 2-kernels into two Brauer problems, thus finding the obstructions in some specific cases.

be a finite group extension. The embedding problem K/k G A then consists of determining whether there exists a Galois extension L/k such that K ⊂ L, G ∼ = Gal L/k , and the homomorphism of restriction to K of the automorphisms from G coincides with π. The group A is called the kernel of the embedding problem. If there exists a Galois algebra with the aforementioned properties, then we also talk about "weak" solvability. Given that A is contained in the Frattini subgroup of G, i.e., rank G = rank F , the two terms are equivalent.

Let A be a cyclic group of order m, let ζ ∈ K be a primitive mth root of unity, and denote µ m = ζ ⊂ K * . If F acts on A and µ m in the same way, then the embedding problem K/k G A is called a Brauer problem. We can identify A with µ m and denote by c the 2-coclass of the extension (*) in H 2 F µ m . It is well known (see [Mi2, ILF]) that K/k G A is 355