Embedding Obstructions for the Generalized Quaternion Group

Abstract

In this paper, we examine the obstructions to the solvability of certain embedding problems with the generalized quaternion group over arbitrary fields of characteristic not 2. First we consider the Galois embedding problem with abelian kernel in cohomological terms. Then we proceed with a number of examples in order to illustrate the role of the properties of the base field on the solvability of the embedding problem. ᮊ

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

“…(3.6b): The obstruction is −1 −1 ∈ Br k . This is exactly the same result obtained in [MZ,Example 3.4].…”

“…By [ILF,Section 4] the compatibility condition for embedding problems with cyclic kernel of order 4 is also sufficient for solvability (see also [MZ,Corollary 3.3]). We define homomorphisms e f , and g from F in +1 −1 by σ a = a e σ σi = i f σ , and g σ = e σ f σ .…”

On the embedding problem of central cyclic extensions of abelian groups

“…Clearly, the problem (K/k, H, µ q ) is Brauer, so from the proof of Theorem 2.1 given in the paper [6] it follows that H 1 (Ω k , Hom(V, k× )) = 0. Hence the homomorphism γ :…”

On Galois cohomology and realizability of 2-groups as Galois groups

“…Corollary 2.5. ( [MZ,Theorem 3.2]) Let the kernel A be abelian and let ρ χ = χ ±1 for all χ ∈Â, ρ ∈ F . Then the compatibility condition is necessary and sufficient for the weak solvability of the embedding problem (K/k, G, A).…”

“…Corollary 2.6. ( [ILF,§3.4.1], [MZ,Corollary 3.3]) The embedding problem (K/k, G, A) with a kernel A isomorphic to the cyclic group of order 4 is weakly solvable if and only if the compatibility condition is satisfied.…”